I have a HQ 16 on a frame. With that in mind,
1) my bobbin winds to full and stops. There is no adjustment to the bobbin winder that makes the winding stop earlier - at least I haven't found one. Stop button works well.
2) It is very full when it stops on its own, but not overfull. It should never extend beyond the edges of the bobbin.
3) Not sure what squishy means, but if it worked well you're ok. I wind the bobbin on a medium speed, which gives a firmly wound bobbin. When I wound them slower, the bobbin filled so loosely that I couldn't use it. Never have wound a bobbin at top speed, so can't say what the results would be.
4) Rattling when you near the end of the bobbin is normal. As you discovered, it warns you that you are about to run out of bobbin thread. That's a good thing!
5) Leftover thread on the bobbin: Try putting the bobbin on your dsm thread holder and use the leftover bobbin thread to piece another quilt or for the binding. I have even wound it onto an "L" size bobbin to use when piecing or binding.
6) Unless you are using pre-wound or cardboard bobbins, you can wind new thread on a bobbin when it is empty. As always, price of thread depends on the particular thread you want to use.
Have lots of fun quilting!