You don't do anything to switch them out, the people change daily but it'll always be either the Gym Leaders or a few of the people that you've teamed up with along your journey like Cheryl or Riley or even Buck for example.
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As you progress in the game, people will give you apps for your poke tech. they can show maps, caught Pokemon, time, drawing pad and more. to switch apps press the button on the touch screen on the right. yay!
you can't it only goes up to gear two. To switch gears press the B button.
Actually you just have to press view your pokemon team then click switch your pokemon team in the top, then when you get there click whatever category normal, shiny etc. then click what pokemon you want to switch.
There are many Pokemon that you can have. You just have to switch them out in the PC when you want to.
mesprit is a wandering Pokemon, so after it runs away, it goes somewhere out in sinnoh. if you switch locations at certain places, mesprit will eventually get to where you are. do not fly to a nearby location with mesprit, or it will run away to a different place. hope this helps =D