A bugs/problems forum is a tricky one because any of the things in the other forums might or might not be bugs, and most of the posts here are because of problems! That's why we split it down roughly by program area, but we're not too worried if things end up in the wrong place.
This is a known bug that I've reported at least twice -- the picture shown in the Navigator is different than the selected photo while working in the Develop module. Switching modules to Library and back to Develop doesn't fix it, but mousing over filmstrip does. I have never found a way to reliably reproduce it. Can you reproduce it at will?
Do you mean like this, see screen captures. If you right click in the filmstrip area you can have a different thumbnail display in the Navigator when you mouse over. Just have to be careful where you leave your mouse pointer.
I can, but not consistently. If I quickly move the cursor from the filmstrip to the main Library or Develop image, sometimes the wrong photo continues to stay in the Navigator panel. I suspect this is related to the bug where the double-headed arrow gets stuck as the mouse cursor. I find it easier to do if I toggle off the toolbar, but that isn't necessary.
Same problem, just a bit of mousing over does the trick ! I also had a weird cropping issue whereby it snapped to one of my presets as I was adjusting the crop. I have also experience many other weird issues that come and go but are not really a huge enough problem to get upset with. But one thing I have noticed is that theses oddities seem to happen after LR has been open for a while and I have undertaken quite a lot of work...never at the start of a session. A close down and re-start always seems to solve these little issues.