I think he would rather die in a noble battle than flee like a coward to Constantinople.
That one seems easy to me: there's nothing for him to escape to. He made his name as a military commander of non-Christian state's military. Now he's been humiliated by Timur and he's been around a long time, probably pretty old and concerned about his legacy. So dying bravely might have appeal than living as a man without an army who was just one more person out-classed by Tamerlane.
I suppose so, but surely, that's having Michael behave more like a stereotypical comic book hero than a real human being? Were he under different circumstances then I could possibly agree with you, but this is a man who's wife has just delivered him twin baby sons, it's hardly as if he has nothing left to live for. Obviously this is KP's TL, and I don't like to criticise what has been a very fine work of AH so far, but this particular scene didn't make a lot of sense to me.