why was the wedding (R/T) quick and quiet
Well, Remus says Tonks' parents don't actually like him much, right? So given that even the parents are disapproving, its a time of war and only the couple are happy about the prospect of marriage, then I'd guess the marriage would be quiet.

why did Voldemort ask about Draco babysitting the "pups"
You-Know-Who was simply jeering on the fact that a relation to the Malfoy family married a werewolf. It'd be considered a sin as far as he is considered. Draco was merely being used a way to the mock the family, I guess.

why Tonks was glowing and Remus was miserable at Bill and Fleur's wedding?
Well, to this I can't be really sure. Could it be the fact that Tonks was pregnant at that time or is it too early still? Maybe she was just happy. Either of them, anyways. Someone else' comment could be useful, I think.

As far as Neville's question is concerned, I'm inclined to believe both Michelle and Alice. Given how Remus behaves towards the news of his own child, I'd guess that he'll consider it harmful for Sirius' son to be any close to him. However, then again, would he just leave his best friend's son like that? Alice is right because Dumbledore himself took the charge of Harry's caretaking and Remus is likely to have respected his decision to bring up Harry without any knowledge of the Wizarding world.

On a second note, see the difference b/w Remus and Sirius. I know there is no way we can be sure if Remus ever tried to lay his eyes on Harry before he they met on the train, but see how Sirius first comes to Privet Drive before moving towards Hogwarts. Now, I'm wondering that if Dumbledore had not offered him the job, would he have approached Harry at all? Hmm...