I mean, if Remus was helping Sirius, Snape's idea of putting him under closer surveillance would have served to put him on alert, while the assumption that he was totally trusted would keep him calmer... and doing whatever he was doing.

Absolutely. I think the implication here is that Dumbledore knew all along (or at since since before the beginning of PoA) that Sirius was innocent. I'm a big Ron = Dumbledore proponent myself, so both his knowledge and his noninterference make sense to me.

And Voldemort is very, very powerful, but in the killing scene, he'd just killed James. Maybe if he had killed Lily, he wouldn't have had the strength left to kill his target, Harry?

That makes sense. From a dramatic standpoint, Voldemort associating Lily with his own mother always worked for me... (and apparently, this is what happened, if we believe Dumbly's explanation at the end of OotP). But on the other hand, I like to think that he is indeed a fairly cool evil mastermind once he gets his stride, not to be tripped up but emotion. In the actual books, we see him struggling up from nothing, but in the early 80s he had everything going for him.