Yes they can! Dobie at one point was my newest villager. I even moved him in by amiibo. He was the first villager to ask to leave though. I let him go, but I'm reconsidering moving him back in because I kinda miss him.
Yes, they can still ask to leave whenever they feel like it, even on the same day that they finish moving in if you're lucky (or unlucky, if it's someone you want to keep). I mentioned it in another thread last month that Lucky wanted to move out the same day he finished moving in. Sure, he wasn't my 10th villager (he was my 9th at that time), but my town already had 10 villagers (Roald was unpacking) in it by the time he finished moving in.
The only other villager that has asked me to move out was Agnes a few days before her birthday, though she was one of the 2 starter villagers for me. I guess Nintendo might have wanted to make the villagers fickle/unpredictable when it comes to asking to leave your town.