Yes, you can. Tons of people have had to do this because their villagers unexpectedly began packing up and needed someone to hold them until they were ready to move again.
...I don't think they still recognize you, though.
Yes and no. You need to cycle through 16 villagers to get them back once you have traded them. So let's say you trade a villager, you'll have to trade 16 more villagers to get them back if they do ask to move. It also does not matter where you get the villager from, friend or stranger.
