First of all, no, you can not play as Diddy during single player, he will automatically jump on Donkey Kong's back when you find him. Only the second player when playing with someone else can be Diddy! And no you can not use the classic controller!
Wow... I don't know what to do now, Diddy has been my favourite DK character and one of my top 3 Nintendo characters since 1994.
I don't have anyone to play with so I don't know if I really want to get the game now. Can't he even be "unlocked" at the end or something? I might just wait for DKCR2 then, or maybe get a new girlfriend (although I'm still not over my breakup), or just get one of those pay-to-play girls that I saw a few months ago here on GAF, oh I think those were just for online play.