There is an in-game hint for the second paperclip, I think it's called "Trades in the Sky?" and is held by the Canyon hint giver.
The key with an eye opens a secret entrance somewhere. It doesn't look like a normal door, so that's probably why you didn't notice it. I can give more information if you can't find where to use it.
If the carrot is in your bag, Bunneys will loosely attempt to follow you. They don't cross screen boundaries though. I think the only other use is that it counts as a "bunney-related item", which means it can be traded to a person who wants such things.