OK, so I guess the Doctor really is immortal then? Granted, he/she was never actually going to die and we only have the Master's word for it that any of what he said was true (the Matrix has lied before - also with the Master's involvement, in fact) and there's still the possibility that instead of a pre-Season 1 Doctor we have a Season 6b situation (that I'd prefer - it explains why Ruth's TARDIS is also a Police Box, for one thing).
Would the Cybermen really confirm whether somebody really was a Cyberman by sight? You'd think they'd have some sort of wireless confirmation. Then again, you'd also think they'd have some sort of targeting system so they can hit their targets occassionally! Though I did like that the Cyber-Lords had their traditional Time Lord headdresses - evidently the Master is a fan.
Overall though, one of the better finales and we even end on a good old fashioned cliffhanger! Also, kinda liked that The Master would be OK with The Doctor killing him. I think there is often along with the meglomaniacal aspect to his personality (or personalities) a desire for self destruction (which might account for how many times he's died!*)
Not all Gallifreyans are Time Lords (as was actually mentioned in this episode) - in fact, not all Time Lords are Gallifreyan (eg River Song). If what was revealed is taken as true (and as I said, I have my doubts - especially if a future Showrunner wants to RetCon it!) then The Doctor's Genes for Regenration (and possibly double heartedness) were spliced into the Time Lord genetic code by Tecteun.
I doubt it. The Doctor has always been prepared to sacrifice him/herself. I think she doesn't like the idea of killing what was left of her people (until they next come back anyway), however much she isn't a fan of Time Lord culture in general. And The Master has always been portrayed as a Frenemy, who'd be dying with the rest of them.
ETA: And did we really need to rename the CIA (Celestial Intervention Agency, of course)? Or is this a UNIT/Torchwood (or Section 31/Starfleet Intelligence, to switch fandoms) situation where there are two organisations doing essentially the same job? I can't believe the real CIA (Central Intelligence Agency) would complain about how they're portrayed by a BBC show (although I wouldn't have thought the UN would complain about UNIT, but allegedly they did).
* And this time I'm sure it will be permanent, honest!