No, it is not possible to master all styles. Or really even learn them. There are over 400 Kung Fu styles alone, about a dozen major karate systems, several different schools of Aikido and Jujitsu... Most people are lucky if they "master" one system after decades of study.
The "Shaolin Monks" these days are barely martial artists. The Chinese government killed or exiled all the authentic Buddhists in the 1950's, and the current "monks" aren't actually monks at all. They're famous for performing extraordinary physical, acrobatic feats; they're also famous for not having too much actual practical fighting skills, either.
Mixed Martial Arts is not a fighting system, it is a training and competition format. Fighters in MMA come from a variety of backgrounds and train in a number of martial arts. For example, Georges St-Pierre has a black belt in Kyokushin karate and Brazilian Jiujitsu, as well as extensive training in boxing, Freestyle wrestling, and Muay Thai; Anderson Silva has black belts in Taekwondo, Judo, and Brazilian Jiujitsu, as well as training and ranks in Hapkido, Capoeira, boxing, and Muay Thai. So you can enter MMA competition or use MMA as a training format, no matter what martial art you come from.
Wanting your teacher to be Asian is a little weird, I've gotta say. None of mine have been Asian and I've learned martial arts just fine. Are you sure you're not just watching too many movies?
With all that said, the good news is that you can learn multiple martial arts over the course of your life, including the use of the Japanese sword (Kenjutsu). The best way to start is to look in your local area for a teacher. I would suggest focusing on one martial art at first, until you're about brown/black belt level (four or five years). That will give you a point of reference from which you can start examining other martial arts.