[ SO SHE CAN BE THE VERY BEST, LIKE NO ONE EVER WAS. TO CATCH THEM IS HER REAL TEST, TO TRAIN THEM IS HER CAUUUUUUUUUUSE. ]

Not if you don't want to...you can do whatever you want to do, really. There's two type of trainers in terms of ID cards..."trainers" and "breeders". If you want, you could try to learn about Pokémon nursing, maybe...?

[ Well, what Nanami names them... ]

I name mine after video games and video game characters, because I like those names. So...whatever you like, I think.