Amarok,
Please note this is my best solution at this time and I’m yet to robustly test it!
first question: I’m having the same problem you are with my g40 hats. I think i’ve fixed it though:
Switches that have multiple modes of operation (long presses etc) cannot be combo keys (so, ‘Shift-Delete’ for example). As far as I can tell, this is TMS and the comms switch.
So, what you need to do is put the comms switch and TMS to the left/right/up/down and del/end/pgdn/home key binds in BMS and swap whatever was already on those (ICP for the arrow keys) to a ‘shifted variation’.
Then, assign keystrokes to the hats (in the Logitech software, sorry Amarok its the only way) to do the left/right/up/down and del/end/pgdn/home keys and you should be good to go.
Second question: I haven’t seen that (i’ve experienced the R1/2 on the throttle not working and fixed that with some rewiring). might be worth cracking open the joystick and seeing if your have stressed wires in there.