Lighting – How to open this flush mounted kitchen light fixture

I had a very similar light, and the key was this:

Any upward pressure exerted from grabbing the glass created friction that made it not turn.

You want to touch as lightly as you can, near the edges, and try to apply rotational energy only, with no pushing up.

Unfortunately, that is roughly impossible to do if you're reaching so high that you need to press up on the glass to maintain balance. So, you still need to solve the problem of getting higher (and you should NOT climb on anything placed on a bed unless it's highly stable). But if you can get a bit closer, try less pressure, rather than more.

It feels very counter intuitive, but it helped me in a similar circumstance.

One other thing that may help, suggested by none other than @Joel Spolsky, was to try using rubber or latex gloves to allow you to produce more friction without more pressure.

These type of boxes usually have 1 or 2 screws going through the bottom into a steel bar that spans between ceiling joists.

I think you are correct that this box is not properly supported.

If you have access from above, you can put a support (2x lumber) across to each joist and then screw through the box into the support.

Options from below only would involve removing the existing box and using one of several replacement boxes. Removing the existing box will involve turning off the power and prying the electrical cable loose from its strain relief. From the looks of the box, it should pull down. Fixture weight will determine which one to use.

The heaviest duty box is one rated for ceiling fans. It uses expandable threaded supports that wedge against the joists side to side. This box would be rated up to 50lbs.

An old work round plastic box has three retractable wings that clamp the box against the drywall. This box is suitable for lighter fixtures, perhaps up to 10lbs. It is very dependant on intact and unbroken drywall for best performance.