Do you like Theodore? Personally I like Theo, but dislike Theodore, so I wouldn't - and I'm someone who would always put the full versions of Ollie/Freddie/Alfie et al on BC. Theo seems like a proper name to me though, so fine on BC.
No, not unless you like the name. Theo is a perfectly sensible and proper name. I've a short name as my full name and have never wanted another option, my name is my name. I'm not going to start calling myself something different!
I love Theo, such a great name, personally I don't like Theodore, so just Theo I would say. I am sorry but I can't stand Teddy or Bear, but that is just me. It's a little bit like the name Charlie, that is their name, not many are Charles on BC in my experience.