If it is synced for a while and just shuts off, try a new set of batteries. The wii, however, should tell you when batteries are running low though.
To the 2nd question, just try a new set of batteries and see what happens. Also turn the other controller off, then try and turn the one on so it is the 1st player and see what happens.
Sorry if I didn't give the best answers, but it's worth a shot. Let us know what happens!