This essay is not going to be as long as you think.
In broad terms there are two ways to approach the topic of time travel in Doctor Who. In the top-down approach, I'm going to start from a strictly external, real-world point of view, looking at how the needs of the television production shape the time science inside it. In the bottom-up approach I'll look at the "in-universe" evidence and see if it's possible to assemble that evidence into anything meaningful and/or internally consistent.
Doctor Who is a television show which has existed in one form or another - with one significant gap - for more than half a century. The show has been running for so long, through the televisual equivalent of several geological epochs, that it's impossible for the show's purpose and driving forces not to have changed dramatically over time.
I'm not a scholar of the early days of Doctor Who but there must have been an early period of uncertainty when nobody really knew how much potential the show had, or whether anybody was going to watch it; a time when none of the show's mythology had been really explored and the rules hadn't been laid down. Some time after this, a time must have come when it was realised that Doctor Who was a television programme with serious staying power. As to whether the show was carefully designed from first principles to be this way, I couldn't say, but it cannot be denied that, whether accidentally or not, the entire show is built for longevity.
Two central story elements support this. The TARDIS, which is able to travel anywhere in time and space, is a "story spinner" of unlimited power, right out of the textbook. And the Doctor, the lead actor and main character of the show, is able to "regenerate" and change his face while remaining the same person, providing the show with a continuous thread of continuity across decades of stories, regardless of casting concerns. These are the two narrative powerhouses which more or less render the show unkillable by conventional means. It's a really smart design, and everything else - Time Lords, companions, intergalactic peril, monsters of the week - is built around this core. Doctor Who is a show which can theoretically last forever and - Batman take note - never even need a reboot.
Once this was realised it must also have become clear that this timelessness and (long, long) continuity was valuable. Extended and convoluted continuity isn't an unambiguously positive thing for a franchise to have, because the franchise can very easily become murky and inaccessible. But it certainly has merits and, because such continuity is time-consuming to build up, these merits are relatively rare and worth hanging on to. Because Doctor Who is a show which has run so long, and because it has the potential to run for so much longer after the present day, it becomes desirable to keep things that way.
Under these circumstances, the decisions you make in your storytelling - including your portrayal of time travel - change. It doesn't matter which particular episode of Doctor Who you are writing today, and it doesn't matter what year it is-- you can be sure that this isn't the last episode of Doctor Who which will ever be written, or the last series, or the last Doctor, or the last generation of viewers. The show may die and the show may come back again and the Doctor will still be the same character. Fifty more years from now, there's a genuine chance that people will be coming back and watching your fifty-year-old episode and comparing it with recent ones.
More to the point, there's a good chance that this time next year you will need to write another episode. Maybe a whole series of them!
Under these circumstances, what you obviously want to do, above all else, is avoid closing the door to future stories. This is a completely different proposition from writing a single book or a film. Those are fixed, capped stories, where you would have relatively unilateral control. You could kill everybody off by the final page, and as long as the story itself is compelling nobody will hold that against you. But writing for Doctor Who is different. This is franchise fiction, a renewable resource, a golden egg-laying goose. You must not kill it!
You also want to avoid loudly contradicting previously-established history. But luckily for you, nearly all prior writers were in exactly the same position as you are now. Hopefully, they won't have done anything for you which makes your job impossible, and hopefully you can be relied upon to pay this favour forward and not make future writers' jobs impossible either.
So the way this works out in practice is that you reveal the bare minimum of necessary hard facts to support the story currently being told. You keep information back, you wave your hands. You do acknowledge the important questions, but you nearly always dodge them, by ignoring them, or lying, or giving ambiguous answers. Locking down any specific fact not only constrains all possible future stories which could be told in that area, it also instantly rules out an unlimited number of other possible facts which could have been selected instead. Even seemingly harmless little factoids can turn out to be massively inconvenient years down the line. (Smarter writers are able to work in the opposite direction, using hints to bury groundwork which later stories can, if they choose, pick up and expand on.)
This can be seen in nearly every episode of the show. You only have to look at the glacially slow pace at which new information about the Doctor's species, homeworld, people, backstory and name have been revealed. What is the Doctor's name? What were his parents like? Is he married, does he have children? How many regenerations does he have? Is he always male, is he always white? If not, then why is he? Whose TARDIS did he steal and do they want it back? How long does a TARDIS live, how do you make one, where do they come from?
And obviously, as a natural result of this requirement to be cautious with the truth, the Doctor must be a private, secretive man, not liable to blab about his personal details. This remains a constant regardless of his external personality, which is frequently brash and noisy. Companions do often ask him the important questions directly, because it would be bizarrely inhuman of them not to, but he always sidesteps them somehow.
"Rule One: The Doctor lies." Do you see how unbelievably smart this line is? It means even on the rare occasions when the Doctor apparently talks candidly, we can still roll all his claims back later on!
That's not even the real Rule One!
With this in mind some things become naturally obvious about how time travel must work in Doctor Who. The keyword here is possibilities.
Time travel must be possible.
Time travel must be possible in both directions, forwards and backward.
Time travel must be relatively straightforward most of the time. (Contrast with e.g. Back To The Future where it is extremely difficult to travel through time, and this difficulty forms the principal conflict of the (finite) story.)
That's not to say that we can't build stories where time travel is roadblocked in some way. But we want the flexibility to tell stories which aren't specifically about time travel, where we arrive in 1941 at the beginning of the story and leave 1941 at the end without any particular issues. If, every week, we need to find some outstandingly rare MacGuffin to power the time machine, then that's going to get quite tiresome quite quickly. As well as quickly straining credulity.
This gives us the TARDIS's personality. It's reliable to a fault when the plot demands it. It's finicky and broken when the plot demands it. Making the TARDIS work is sometimes a critical problem and sometimes a non-issue.
Note that this kind of fuzzy "Uh, it depends" behaviour is extremely honest. It works for us. Because this is how real people and animals and machines behave in reality! If something worked the same way every single time for a hundred years, that would be freakily suspicious.
History cannot be fixed.
If every single time travel story boiled down to a collection of closed causal loops, that would get dull and predictable and repetitive. Furthermore, characters would be able to abuse this knowledge to solve problems too easily. That's not to say that causal loops should be outlawed entirely, because they're powerful and fun in the right hands. But other formations must also be possible. We need to be able to have it either way when we need it.
Plus, there must be a credible danger of altering history, otherwise there is almost no tension in historical stories. "The Daleks are making a bid to conquer Earth in the eleventh century? Well, that obviously isn't going to work out for them. So... Bye!"
If history changes, then somebody should notice. The Doctor (and possibly his companions) should be able to perceive these changes, probably by virtue of their privileged perspective as time travellers.
"Let's just hop in the time machine, go back in time and fix everything" must not be a viable catch-all resolution to every episode's conflict. Time machines are insanely powerful devices for breaking stories and we have to limit that power one way or another!
The rules of time travel must be totally internally consistent within the confines a single episode. This is because fiction needs to make sense. (No, "It's science fiction!" is not justification to spew out complete narrative gibberish. A story must have rules, and it must obey those rules.) Ideally the rules should be consistent across the whole series, but this is nowhere near as critical, or practical.
And other than that... pretty much anything goes.
These rules give rise, on different days, to stories where time travel is nothing to do with the story, stories where time travel is involved and critically important and super difficult, and stories where we bounce backwards and forwards in time for comic effect. It means most things which make sense and drive a story are possible, and most things which make too much sense and would kill a story are impossible.
It also gives rise to some fairly obvious... I don't want to say contradictions (or paradoxes - a paradox is a scenario which apparently involves a contradiction but does not)... but at least I have to call them pain points.
For example, the characters can't fix everything using simple time travel because once they arrive in a situation they are "part of events". What does that mean? How long do they have to be part of events before they can leave? Is it like parking, no return within thirty minutes? Can they come back a hundred subjective years later? Can a different Doctor come back and fix this? How far back in time can you go before you're no longer part of the same events? What about in space? If 1930s New York is "blocked out" in some way, what's wrong with going to 1920s New Jersey and hanging around for a while, then getting on a bus?
There are fixed points in time. What if someone destroys the universe? What happens to all the fixed points after that time? Does the universe cobble something together somehow?
It doesn't matter. Well, it kind of matters, because it's never terribly satisfying for one character in the story to effectively turn to camera and say "We just can't fix this situation by doing the obvious thing, and I can't explain why! You just have to believe me that there's a genuine conflict going on here!" but as a show Doctor Who does tend to survive.
It does this principally through the means of excellent casting. A good Doctor can sell it. And regardless of the strength or weakness of the material, every Doctor (in the revival, anyway, can't speak about the classic series) has been able to sell it. And so we buy it, at least for that one episode, and that's as long as the illusion needs to last.
Still... Play this card too many times, leave too many obvious questions unanswered, and the wheels start churning and dissatisfaction does start to set in. "Don't think about it" just doesn't work as a defence mechanism. You cannot watch the same show for decades and decades without, at some point, starting to think about it.
And so it becomes somewhat desirable for the creators to try to maintain some kind of consistency, and for the viewers to try to construct some kind of framework.
This is where I get all fan fictiony.
Doctor Who is a show involving more time travel than any other show I can think of. It is contradictory and inexplicable and different every day. But it also provides us with some natural ways to understand all of this. Because Doctor Who is a show which is very often actually about time travel, and often about time, and the nature and structure of time. It even features characters called Time Lords who are - if I understand this right - actually responsible for the maintenance and upkeep of time itself.
This implies that time in Doctor Who is, if not a living thing, then at least a dynamic thing. It implies that the laws of time are subject to deliberate modification. Some or all of the laws may have been installed deliberately. And the laws may also change naturally, over the course of... well, some other form of time.
And that's basically it! It would be insane to take every single episode and try to line up all of the evidence. And it would be nearly impossible, and it isn't necessary. Genuinely malleable laws are all you need to justify most of the phenomena seen in the show. Time travel in Doctor Who is just different every time. The laws vary with real time, with subjective time, from place to place and from person to person. The laws will differ for two Doctors visiting the same era on the same planet. There are legions of edge cases and exceptions. There are freakish phenomena and monsters crawling out of the walls.
In-universe, although there may be an overall theory which explains everything, it must be insanely complicated, and to try to explain it to a doddering human viewer would be futile. And understanding time in terms of those laws is about as useful as understanding water - placid lakes, whitewater rapids, oceans, icebergs - exclusively by studying fluid dynamics equations. Eventually you might get somewhere, but you can't learn to surf that way.
Fortunately for us, the Doctor has an instinctive understanding of all of this. He has a perspective which lets him intuitively understand the fabric of local time, and what is or is not allowable. (This drives him a bit crazy sometimes.) He can explain what's going on to the companion (read: viewer surrogate), in simple terms, and always does. So we don't need the full and gory details. Which is fortunate, because honestly they simply do not exist.
And, as I say, it's probably for the best that they don't.