1. How did Boe know? About the Doctor not being alone, and about the fact that their encounter in Gridlock would be their only one after New Earth, and their last? Does this relate to the mystery of how old the Face of Boe is? Millions, even billions of years - impossible or not? Can the Face of Boe travel in time? Had he been to the End of Time? Or is it simply that he can 'see' to both ends of time (or at least the future) in some kind of omniscient fashion? He certainly seemed to be able to send the Doctor a psychic message without the Doctor needing to be in the same time-period as him. Either way, there is a lot we still need to know about him.
2. After Professor Yana (ooh, that name was a nice touch, wasn't it?) has opened the watch and become the Master again, he removes a circuit-board from his navigation screen thingummy, and scoffs about the 'Utopia' dream. To me, this made it pretty clear that although there might once have been a real Utopia project, that signal didn't come from it. It must have been set up by the Master before he transformed himself into a human, to 'trick' his human self, and the other humans on Chantho's home world (Malcassairo, it's called, apparently), into believing in it and trying to get there. To me, two subgroups of rather important questions spring from this:
2a. Was the Professor really found on the shores of the Silver Devastation (the Face of Boe's home world - if it is a 'world' as such) as a naked child, or are those false memories like John Smith's parents? Because of the false signal, my working assumption is that the memories, too, are false - he arrived on Malcassairo at the End of Time still as a timelord, infiltrated the group of humans living there, set up the signal, and then transformed himself. Presumably, this was about 17 years before the Doctor and co arrive, since that's how long Chantho says she's been with him. Given all this, was the reference to the Silver Devastation just intended to remind us viewers of old Boe, or was it a clever in-story clue left by the Master to himself?
2b. Again, assuming that the signal was false and deliberately set up by the Master - WHY? Was it just to give himself something to do as the Professor? Perhaps to prevent himself thinking too hard about his pocket watch while the Time War was still raging and it would have been dangerous to open it? Also, what's going to happen to all those poor people on the rocket now? Are they blasting off to the middle of nowhere? Or was there something where the false signal was coming from - just not the Utopia project, as everyone thought? If so, was that something in some way connected with the Master's plans?
3. What actually was the Master's long-term plan? Presumably at some point he'd hoped to return to his Time Lord self, and to eras in which he could Reign Supreme once more (ideally with all the other Time Lords, including the Doctor, having been wiped out by the war in the meantime). How did he intend to make this happen? Surely he couldn't have banked on the Doctor and his TARDIS turning up on Malcassairo, even if he might have guessed that the Doctor was strong enough to survive the war. Would the launch of the rocket - a project which he knew would take a good long time after his arrival on Malcassairo - have triggered it? What if he'd died while still Professor Yana? Would that, too, trigger a return to Time Lord form?
4. Somewhat related to the above - where is the Master's TARDIS? It seems the most logical way for him to get to Malcassairo - so is it still there? If it is, I'm betting the Doctor will be able to detect it using his Sonic Screwdriver, no matter what shape it's in.
5. Will that be how the Doctor, Jack and Martha get back to Earth? It needn't be, as angelich has pointed out the alternative route offered by Jack's carefully-explained watch.
6. I'm not in all honesty sure I care about this, but it has to be asked - how / when did the Master escape from the Eye of Harmony in the TV movie? The most likely theory for the moment seems to be that he was taken out by the Time Lords to help them in the war, but then did a runner.
7. Why drums? Someone somewhere on doctorwho (can't find the post now) suggested that it was a memory of his second heartbeat as the Master. I really like that theory, as the sound effect we heard did include something that sounded distinctly like a heartbeat, as well as actual drums. Until it became clear that he was a hidden Time Lord, I assumed that noise was just his own, human heart, beating loudly as they sometimes do when we have a funny turn. But as soon as I saw that theory, it made sense. Doubtless we'll learn more next week, anyway, given the episode title.
8. How did the Master manage to regenerate? (A question which is of course particularly important for its implications about the Doctor's future). Has he got himself a new stock of regenerations from somewhere? Does it have anything to do with his time in the Eye of Harmony? Or how he got out of it? I don't, for the record, think it had anything to do with the Doctor's hand - there was absolutely nothing at all in the episode to suggest that. My assumption is that he took the hand into the TARDIS to somehow fool it that its rightful owner was present, and thus allow him to fly it successfully - and of course if that was necessary, logically the Doctor won't be able to use the Master's TARDIS to chase him, since he doesn't have any of the necessary DNA. It'll have to be Jack's watch.
9. The conversations between the Doctor and Jack seemed to me to be suggesting that the Doctor knew perfectly well that Jack had become immortal in the year 200,100, and actively chose to leave him behind on Satellite 5 for that reason (his 'Wrong'ness) - not just in Cardiff when he was running towards the TARDIS. If this is so, how did he know? I really don't remember any sign from The Parting of the Ways that the Doctor knew Jack had come back to life (or indeed been killed in the first place) - as far as I understood until I heard their conversations, the first time he'd set eyes on Jack since Jack went off to fight the Daleks was in Cardiff. Maybe this is just messy plotting - I don't know.
10. Anyway, why does Jack still so desperately want to see the Doctor again? He says that he originally travelled back to Earth in 1869 using his watch, and then realised that he was immortal after a fight in 1892. He's had over 100 years to come to terms with being immortal, then - and he also explicitly says that he doesn't want to die. So what can the Doctor do for him that nobody else can? Is it just get him off 20th / 21st century Earth? Make him a Time Lord, maybe? Or something else mysterious that we haven't thought of yet? (I'm ruling out "He just thinks he's that hot" as unlikely in a children's programme). Whatever it is, it seems it will be resolved to the extent that Jack is free to (and wants to) go back to Torchwooding by the end of the current series, since there's to be a second series of Torchwood, and by all appearances John Barrowman will be in it.
Er, yes. That was quite a lot of questions, wasn't it? It's good - I'm done now.