When the Doctor shot the white-point star and broke the "link", why is it that he as the only Time Lord to not get sent back into the Time War period, and why was the Master sent back if the Doctor wasn't?
tl;dr Rassilon had punched a "tunnel" through the Time Lock from Gallifrey to Earth, which The Master simply chose to walk into. He was not "sent back" for any reason other than he chose to chase after Rassilon. The Doctor didn't follow them back, so he remained outside.
To start with, we know very little about the mechanics of that link, and in particular, we don't know how exactly Rassilon managed to break through a Time Lock from the inside. This seems to be directly counter to the point of the Lock, though it's not the first time we've seen someone pass through the lock and come back out (e.g. Davros). From what we can tell, the process worked something like this:
- During the final days of the Time War, the Time Lords received word that The Doctor had obtained "The Moment" (which we know nothing else about), and were pretty sure he was going to use it to place a Time Lock around the entire war.
- Rassilon was looking for a way to escape from this Time Lock, and hit upon the idea to embed some kind of link in The Master's mind. Since the Time Lock was not actually up yet, he was able to use the Scism (which all Time Lords visit) to implant a signal into The Master's mind.
- The Master subsequently fled the Time War to the end of the universe and used his Chameleon Arch to become Prof. Yana, thus he was not trapped inside the Time Lock.
- Fast forward many years and The Master is now on Earth. Meanwhile, Rassilon is now stuck behind the Time Lock (we have no idea how much time has passed in his mind) and decides to break free.
- Rassilon takes the diamond from his staff, which is locked in the Time Lock, and sends uses his connection to The Master to effectively "tunnel through" the Time Lock to Earth. At this point, it seems like the tunnel is much to small to be practically useful (e.g. it may take a lot of energy to punch through from inside the Lock.)
- The Master hooks up the white point star, which by all rights ought to be inside the Time Lock, and uses it to blow open the tunnel large enough to let everything through.
At this point, everything that was held behind the Time Lock starts to leak out, starting with Rassilon and the Time Lord council. The Doctor/The Master/Rassilon have their little cat fight, and The Doctor destroys one half of the link that is holding the tunnel open. That causes the tunnel to begin to shrink, because there's nothing holding it open anymore. The things that were still trying to come through out of the Time Lock find themselves stuck again.
It's not clear to me if they were "fully out" and "pulled back in", or if everything we saw was still "in the tunnel" and never made it out, but my instinct says that it is the latter. Rassilon is still trying to escape the Time Lock while he has the chance, and it is only because The Master pushes him backward into the Time Lock that he fails. The Master is following the same tunnel in the other direction, into the Time War, even though he was not originally trapped there. Had he not done so, presumably Rassilon would have had a chance to escape before the tunnel collapsed, and would be permanently outside of the Time Lock.
The Doctor simply didn't follow The Master into the tunnel, so he remained outside the Time Lock when the tunnel collapsed.