Trainermon was the shift in pokemon deck building that took the focus away from the pokemon and moved it onto the trainer selection and combonation. At least that's how I've always heard it explained.
Slowking is busted, plain and simple. Coin flips for all trainers, and if they fail they return to the top of the opponents deck. so not only is their card not used, it is now between them and another more useful card.
Sneasel takes 2 dark 20 times the the number of YOUR pokemon in play, not all pokemon in play.
Every so often a thread gets started about unlimited cards being banned and such. Birthday pikachu is legal in unlimited, however "_____'s Pikachu" counts as a card name. So does "Bill's Pikachu" so if you really wanted to, you can run as many as you want, if you have no more than 4 of a peticular card name.
Ancient Mew is not legal since it is not an official pokemon playing card. It is a promotional card, same as the oversized cards that bounced around for a while. Someone did make up a translation for it though, not sure where it is on the internet, but I have seen people with the printout.