Sorry, but there is no trading system to trade Mogas with your friend. However, if there's an offer where you can give free Moga (or send a free Moga when purchase a Moga), that's where you can send a Moga to your friend. It just depends when the developers decide to do that.
I don't know any faster way to level up with Double XP. However, you can use Whistles to level your Mogas. It makes your Mogas last longer to battle and gain more EXP than battle out without Whistles.