How does one start being a Pokemon Trainer without a Pokemon Professor?
There are all sorts of ways one can get his/her first Pokémon. In the Black and White manga, Black borrows Pokémon from Bianca's father to have mock battles (I don't remember if he actually used them to catch his first Pokémon, but this shows that trainers could borrow someone else's Pokémon to catch their own). In the anime, episode 3, Ash Catches a Pokémon, Ash is able to catch Caterpie without even battling it, so sometimes it isn't even necessary to have your own Pokémon first. Also in the anime, in episode 4 of the XY Season, A Shockingly Cheeky Friendship!, Clemont catches Dedenne for Bonny and says he'll hold on to it until she is old enough to become a Pokémon trainer. 

So, clearly one doesn't need to go to a Pokémon professor to start off. One also doesn't need a Pokémon professor to take care of excess Pokémon. In the anime episode Lost Leader Strategey, it is shown that Paul's brother Reggie takes care of his Pokémon.

Information about the referenced episodes can be found at http://bulbapedia.bulbagarden.net/