You can change Pokemon Trainer's starter from the character select screen by clicking the Pokemon in the top-right corner of the trainer's portrait. Alternatively, as scohe001 mentioned, you can also press Y to swap pokemon.
As for your second question, there is unfortunately no way to disable Pokemon cycling on death. It will always cycle between Squirtle, Ivysaur, and Charizard when you respawn.
That being said, the longer you wait on the floating platform after you respawn, the shorter your invincibility period will be. So if you're trying to switch back to your previous Pokemon, I'd strongly suggest getting back into the game as soon as possible to make use of a longer invincibility period to switch.