Typically, you should be on the ball of your rear foot, for several reasons. First, to move you will generally be pushing off that foot. Second, when you throw a punch with your rear hand, you'll need to pivot that foot sharply to get your hip turned and that is much easier on the ball of the foot. Last, when you hook with your lead hand, the weight needs to shift to your back leg and that foot needs to drop flat at that point. Your front foot should be flat.
It is much easier to slide your feet on concrete or canvas, or anything but carpet, but maybe we should use the word glide instead. It sounds better. That whole bouncing and "on your toes" thing is just wasted energy and wasted motion; you can certainly move fast enough and far enough without it. And, as you noted, it is less stable, and stability is a key to taking punches, throwing them, and defending against them.
I neglected to mention another exercise that helps with balance. It is pretty simple; just move, keeping your feet in proper position and moving as described above, but jabbing constantly. You are teaching yourself that the jab does not affect your balance; that is, there is no, NONE, weight shift when you jab. There is a weight shift when you throw the rear hand, and that is something you'll need to feel to catch on to.