No, either participant can yield. So you'd better be sure your champion really wants you to win your case.
I think anyone can nominate a champion, eg the king has his Kingsguard and can't be expected personally to fight anyone who objects to his rulings. However, the champion has to agree.
Any highborn or knight can choose judgement by the Gods in the form of trial by combat.
Not allowing trial by combat is indicative that you believe that the gods are on the other side.
I think only the defendant can require trial by combat.
As an aside, yes the Mountain could always demand trial by combat if he was brought to justice, as it is assumed that the gods would still cause him to lose if he was guilty. However it would be pretty hard to do so without killing him during his "arrest".
Wikipedia has a pretty good article on the real practice of trial by combatthe real practice of trial by combat in the middle ages (mostly in Germany and Britain).