I think they do remember you only if they come back to your town naturally. I mean by game RNG as a new random villager, not just getting them back again from another town.
So far as I know, they only remember you if they were the original villager to have moved in and out of your town in the first place (such as if a friend is holding him/her for you). If you get the villager through another source, such as the campsite or through trading on a forum, then they won't remember you and will simply act as if they're completely new to your town.
Yes, they do remember you if it's the exact same villager. There was even a really cute post some time ago, about a user's experience with this. They'll also remember you when you visit them in their new town. They don't remember you if it's the same villager but not the exact one you had. For example, if you had Stitches in your town and he moved out, he'd remember you if you had a friend hold that Stitches for you and then you invited him back. He wouldn't remember you if you invited some other Stitches.
It depends. If someone is holding your original for you, and THEY return, YES, they will remember you, but if it's one that never lived there, then no, they won't.
Anyway the result should be de same, shouldn't it? I mean, imagine you've lost a villager you loved. Then you get it back from another town (he/she's NOT the same you lost). In about 1-2 weeks, if you talk to him/her every day and do the favours he/she asks you for, the relationship should end to be the same that you had with that villager before he/she left, right?
The only thing that changes is that, if it was the same villager, he/she would just say something like "hey buddie, I'm back again, I missed you!".