Or even just be able to PLEASE delete/remove the run away pets from your page, so you can't see them

Yes perhaps 5/10 XP like for owls and chat? Or more?

A duelling system that matches that of the old "Potterskolen" could be cool in some way or another. 
I remember notifications about people wanting to duel someone was annoying for some, but perhaps that was because it came in the owlery and not in a "notification" tab? It should of course be possible to set one to "cannot get/accept duelling invitations" if one does not want to duel at all or just in a time frame.

it could come with levels, for wich you gain XP when you win, and some achievement badges for it too.

This really would be lovely! It may be a small thing, but it can be annoying to shift between the pages just to check how far up, the hunger bar is

But then again, the things, or problems, you are mentioning here, is only "valid" so to say, if that is how it is played out IN GAME. Isn't it like saying, under relationships, that you are married to your own user or to your characters bestfriend? 

They would still be there for the 'Get all items' achievements. And you can make the 'buying food' cheaper than the 'eat food at this shop' if that is. But could the 'what is the point of all the..' not be said about most things in the shops that does not have an interact option? Like toys and jewelry? ^^