I tend to believe in doing things the first way. I think it's easier to get something perfect at a low tempo then increase the speed than to play something at a high tempo and work out the bugs there. Make no mistake, playing at the high tempo will make you faster, but if you're missing notes and playing with uneven dynamics, well what's the point of playing fast? If you are playing something at a low tempo and you have it perfect, try upping the speed a bit. If you get to a speed that you can't really handle, instead of playing it there, go back to the highest speed you can play it very well at. I have definitely noticed results with this method. Think of it this way, if a weightlifter can't bench press 200 pounds, does he keep trying to push the bar up even though he can't? Or does he go back to 170 and do reps there?
As for muscle relaxation, same thing. Try playing whatever gets you tense at a low muscle. Pay attention to all your muscles, shoulder, arm, left hand fingers not in use, and ok, jaw. If there is a problem, concentrate on relaxing that muscle at that low tempo. As you up the speed, if you notice that you're slipping back into your old habits, wait a second and work out the kinks at a tempo you're familiar with. However, I've been told that my mouth goes into wierd positions when I play. I know lots of musicians who do this, and if it doesn't affect you're playing, (as shoulder or arm tension can) who cares?