1. Does the size really matter, except for aesthetics? I guess if you were trying to collect the various sizes, like how I catch all the Shinies I see, it would matter.
You could try breeding for the various sizes.
2. I don't know if this has changed for E16, because I'm still playing 15, but the Move Relearner is the person in Onyx Ward looking for the lost key. I think you find the key in Lapis.
3. I'm still on 15, so I don't have an answer for you on this one.