Leave the skins in, after molting the shrimp will often eat the shells to regain some of the minerals they lost when they ditched their undersized shell. Never seen my betta eat any of the shells, but it won't hurt him if he does.
Red Cherry Shrimp can do fine with Bettas. They run smaller than ghost shrimp, so beware they make a more tempting snack for your Betta. In a tank that is well planted with lots of hiding places (I reccomend a moss of some kind at the least) they will do OK though. It also will depend on the temperament of your Betta. My Betta loves to chase my RCS (as a result I don't see them much in the day time), but he is half moon with some serious finnage and never seems to be able to get enough speed going to catch them (the adults at least). RCS will predominately chow down on algae which is a nice bonus in a planted tank, and something that Ghost Shrimp will rarely eat.