Loosely: How do Paradoxes work in Doctor Who?
We see in previous episodes,
That the master has to convert the TARDIS into a paradox machine
To allow a paradox to occur.
We also have the the weeping angels, that feed off of time loops. They are however supposedly,
Poisoned by Rory and Amy jumping off of the building between seeing time-looped Rory form the past/future and suiciding...
Add to this that the Reapers came when,
Rose Tyler saved her dad from being run over. Time is fixed when he dies, later on.
The events of The Angels take Manhatten seem to imply that time can fix itself, erasing the paradox, when it returns the Doctor + companions to the present day. How does this tie up with that region of time being impassable? Hasn't time fixed itself by escaping that closed time-loop?
Answer
tl;dr Like everything else in Doctor Who, it's handled "however we need it to be handled to make a good story". The details get wibbly-wobbly as needed, but it's possible to explain all of those events in a mostly consistent matter. Specifically the universe "tends to" fix paradoxes by forcing the timeline to settle into the most stable state possible.
There are basically three kinds of paradox we tend to see in Doctor Who, and they are handled very differently.
Bootstrap Paradox
Also called an Ontological paradox, we see this all over the place in Doctor Who. The standard examples from modern who is Blink but there are tons of cases where someone knows to do something because they saw themselves doing it. There does not seem to be any problem with these paradoxes existing, so long as they are internally consistent.
Predestination Paradox
The predestination paradox is similar to the bootstrap paradox, but in this case the time traveler is forced to do something because they must have already done it in order to be where they are. There's tons of this in The Big Bang but the real major one in is Fires of Pompeii -- The Doctor has to set off Vesuvius because Vesuvius went off, and if it didn't there wouldn't have been an Earth for him to arrive on in 1963 etc etc.
Again, as long as these are internally consistent, these seem to be OK.
Contradiction Paradox
These are the ones I think you're mostly asking about, variations of the grandfather paradox where someone does something in the past that would make it impossible for them to be in the present anymore, etc. These are really really bad in Doctor Who, though I'm sure I can probably find at least one case where someone got away with it. But in general, these cannot persist, and the universe tries really hard to fix them.
The three cases you mention are the most obvious ones, so we can examine them to see how this works.
In the first one we see, Rose's father not dying would change Rose's life in such a way that she's very unlikely to have met The Doctor, thus creating the paradox. Arguably, we see the same effect in Waters of Mars: a person that should have died, is saved by the Doctor. In both cases, that person does eventually die, and most importantly, before they have a chance to do anything else significant. Thus, the paradox was averted; presumably had Rose's dad not let himself die before, say, releasing a new invention, the fabric of space and time would have started ripping apart or something equally vague and bad-sounding.
In Last of the Time Lords, the TARDIS is being used to create a buffer around the Earth to sustain the paradoxical timeline, in a way that isn't explained (though even Jack seems to recognize it immediately.) However, once that effect is removed, the paradox kicks in, and time completely reverses back to the earliest point where things were still valid -- just before the Toclafane arrived, a year earlier. Everything that happened during that time period was undone. The difference here is that there's no way for anyone to "fix" the paradox normally; you can't "unkill" all the dead humans that the Toclafane killed. So the paradox was essentially fixed into place at that point, and the only way to undo it was to erase everything contradictory that happened.
In Angels Take Manhatten, basically the same thing happens. Once Rory forces a paradox, time gets reset back to the earliest point where a paradox hadn't happened yet. Again, there was no way to undo what Rory did once it was done, so the universe had to step in and handle matter. In this case, the angel's paradox involves sending people back in time, so the events of that whole period had to be erased in order to make things consistent. But it wouldn't be enough to just reset everyone to that time, because there was too much "other stuff" that would be undone as well -- imagine the effect on the rest of the world if the entire history of any major city from 1880 to 1938 was suddenly gone. The least-effort way for the universe to fix every timeline to be as stable as possible was to eliminate the thing that didn't belong -- the angels -- and set everything back to the way it would have been without their meddling. (It's never stated but I can guess that events may have rolled back to just before the first angel "killed" someone.) The destruction of the angels themselves seems to have been a side-effect of the temporal chaos that ensued -- since they feed on time energy, it would be like a human eating a pound of C4 and setting it off.
Conclusion
It appears that contradictory paradoxes cannot persist in Doctor Who, but that time is flexible enough to permit events to correct themselves if possible. Once that no longer becomes an option, the entire paradox is undone, and the universe sets itself back into the most stable and internally self-consistent series of events possible, and everything moves on from there.
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