If you were to have an empty balloon, would it float?

1: Assuming the frame is lighter than air of the same volume as the vacuum inside, yes. It would not have a balloon shape, though.

2: By experience with helium baloons, no way. This source (paywalled) gives ascent speeds of a couple meters per second throughout the entire flight, which is also dependent on the balloon expanding as it rises (which would not happen with a vacuum balloon). Earth's atmosphere around 500 km above ground is pretty much negligible (comparable to the vacuum inside your frame, since slight leaks will occur from the frame). Even if the baloon got to that height with some speed, escape velocity from 500 km above ground is still several kilometers per second, which it could not get to within the atmosphere due to drag. Your baloon will end up floating at some height determined by its density (frame mass over volume of vacuum).

Extremely theoretically, if you were to ignore drag and make your apparatus extremely light, it might leave the Earth gravity well. Effectively, it would gain the potential gravitational energy of air of its volume (at a height difference between ground and upper levels of the atmosphere), and transform it into both potential and kinetic energy. Earth escape velocity is about 11 $frac{km}{s}$. The height difference might be about 10 kilometers - a precise calculation would have to take into account how air density varies with height. This corresponds with a velocity of about $400 frac{m}{s}$ (from the free-fall formula, and assuming a constant gravitational strength, which at these scales is not a problem). That is a speed ratio of ~28, which then gets squared to get the energy ratio. So sufficient energy can be provided for a material of density about 800 times lower than air. Of course, making such a balloon, let alone making it zero-drag, is closely approximated by "impossible".

3: If it were light enough to cross the 100 km "space" boundary, then yes, losing volume would make it go down again. A similar thing happens if a helium balloon pops - it loses volume and falls back on the ground, creating plastic pollution (don't let go of your helium balloons!).

You may also want to read this explanation of the relative sizes of the Earth, its atmoshpere, and space. Basically, atmosphere forms a very thin shell around the Earth and getting out of the atmosphere is not enough to go into orbit or away from Earth's gravity influence.