I'm pretty sure he got shot three times because you gotta kill a dude and what better way to do it than to shoot him a few times. Two to start the regeneration and then another in the middle so he's dead for good.
IIRC the regeneration effect started each time. But either way, I don't think the number three was coincidental.
Also a lot of this stuff seems a lot like Moffat deciding he wants to make more marks on the Doctor Who canon rather than a very well thought out multi-season plan. I GOTTA BE THE ONE WHO HANDWAVES THIS OLD BIT OF CANON AND I'M GONNA DO SO BY INFLATING THE DOCTOR'S POWER LEVEL REGENERATIONS.
I really don't think so. I think handling the regeneration limit is the result of adding Hurt into the line, not the other way around.
Whereas Hurt's inclusion is probably a mix of Eccleston passing on the fiftieth and that it frankly makes for a pretty interesting twist.
I do think that it's quite clear that Moffat's had a seasons-long arc running that involves repeated use of the idea of the Doctor facing his own ultimate end. I don't think the regeneration limit, or Hurt, were originally part of the plan, but I think they both fit nicely.
I also think he's got the good sense to understand that a regeneration is a jumping-on point for new viewers and he should relax the major arc stuff for awhile. I don't think that means he'll try and cram a resolution to every dangling plot thread into the Christmas special, but I do think -- well, I do think that the next season is going to look an awful lot like the last one did, with an episodic bent and no real payoff toward the Silence Will Fall arc until the finale.
I'm fine with this. Last season had the Gunslinger, Cold War, and Nightmare in Silver, none of which really built toward the major arc but all of which I thoroughly enjoyed. Even the episodes that weren't great had enjoyable moments -- Rory's dad, Strax, Richard E Grant.
(And it bears noting that if Moffat were REALLY just doing the War Doctor arc to handle a bit of dangling trivia, Grant would be playing him, not Hurt. Part of me still thinks that's a missed opportunity -- but having Hurt play the Doctor is pretty damn amazing.)