Not really, the balloon would be attached to the top of the bike at let's say 15 meters high and would itself go maybe another 5 or 10 meters up trailing the wire to the bike.
So the rider would still be at the right distance from the ground, it's just that the bike can't fall over anymore because the balloon will help to catch it. You could sit still all day long on that thing (or until the wind caught the balloon...).
The downside of the scheme is the negative feedback loop in the bike once it did became unbalanced. When it's almost balanced it would not take much upwards force to keep it that way but once it starts to become unbalanced the force required will very quickly overpower the balloon.
Or you'd have to make the balloon so large that it would be just incapable of lifting bike + rider off the ground.
The balloon isn't necessary. Inverted pendulums become more stable the taller they get. Think of trying to balance a short pencil on your palm (difficult) versus balancing a broomstick (much easier). Tall bicycles are inherently very stable, and don't need any help with balancing. The difficulty is in climbing that high up to the seat.
I don't think it would be. It'd have to have an upwards buoyant force of most of your body weight to really be effective. At that point we are talking about a huge balloon, meaning it'll cause a lot of air resistance which will cause problems when you're attempting to ride. You'll also lose a lot of traction. I think in the end it'd make it easier to be tipped over (especially if it catches a wind gust), and so would be self-defeating.
This would preclude the balloon trick. :)