I love this problem! I have asked myself this question many times. I think the best way to think about this problem is using forces.

According to physics, objects move (or don't move) because there are forces on them. For example, when you fire an elastic band, you are using a spring force to cause the elastic band to suddenly move. When a ball bounces off the floor, it is using the normal force from the floor to suddenly reverse its momentum (plus spring forces inside the ball--look at this slow-motion video to see what I mean).

When you are standing still, the gravitational force pushing you downwards is exactly balanced by the normal force from the ground pushing you back up, so you don't move at all. So for the school bus problem, if we can keep track of all the important forces on the hovering object, we can figure out how it will move.

First, let's do a simpler case. What forces are there on an object in the bus that is not hovering, say, a student in their seat? As the bus is beginning to move, the student has no forces on them that also makes them move. All they have is gravitational force (down) balanced with normal force (up). But then the seat, part of the moving bus, begins pushing into the student's back. This is also a normal force, but this time it pushes the student forwards. If the bus continues to accelerate, the seat will continue to push the student forwards with a normal force. This way the student "keeps up" with the bus.

Now:what forces are there on an object hovering in the bus? Since it is not falling, it must be supplying some kind of upwards force to balance its downwards gravitational force, and that's it. Nothing is touching the object, so nothing can apply a normal force to it. When the bus moves, the object will be "left behind" by the bus and hit the back window.

The extra part about closing the windows is an interesting addition. You can think of air molecules in the bus like objects, too. But these are very small and moving very fast, so they're constantly zipping around and smashing into each other.

If the windows are open, a current of air might start to flow from the front windows towards the back windows. Now, more air molecules will bump into the hovering object from the front of the bus. This will cause a net force that we call drag, and push the hovering object to the back of the bus even faster.

That depends on how it's hovering and how much air resistance it has. If it's something that isn't falling to the floor of the school bus because the air is keeping it aloft, then it will move along with the bus because the air that is holding it is moving along with the bus. If you have a toy helicopter or something like that, however, that is staying aloft because it has an engine that is giving it lift, then it will hit the back window.