I'm pretty sure they will put what you give them in their homes, unless it's an item like wallpaper. Last year my villagers all put their gifts in their houses and I don't remember them asking about it. If I didn't like their gift though, I just sent them an item in the mail to replace it with.
They do, but only if it's furniture. If they ask for food, clothing, or wallpaper they won't display it in their homes. The ones who requested furniture ended up gifting me the items they replaced so I could've just mailed it back to them if I cared enough.