I bought a Vita at launch and had the same exact issue with it minus the analog stick issue. I thought they would have patched it by now. I don't know because I sold mine a week after because I got tired of it crashing and having to reset it.
Have you tried to hold the power button down for 30secs and messing with the options it gives you. I can't exactly remember what they were. If that doesn't work I would suggest returning it and trying another system. If not well sorry mate, it got nothing else for you other than maybe calling Sony directly.