1 - The main screws on the bottom of the Fighterstick really only hold the bottom. You can remove them without to much concern about losing anything, if I remember correctly (been a while since I took mine apart though.
2 - There is a certain amount of room in there, but I don't know if that is your best option. There are some CH support forums that may come up with better solutions.
Like you, the only thing I really don't like it how light the stick and throttle are. What I did was the following:
I found a heavy board about 1' x 2.5' x 1/2"
I got some large strips of velcro from Wal - Mart and mounted four "female" (furry) strips on the board and 2 strips of the "male" (spikey) strips on each controller bottom. Originally I had about 8 female strips on the board so I could move the controllers around, but now I know what is comfortable so I only have the 4.
I then place the controllers on the board over the appropriate strips.
Also, I have one of those trays for the key board and mouse so when I want to fly, I put the keyboard on the desktop, put the board on the tray (so it is at a comfortable height) and put the mouse on the right side of the board).
This works really well for me and when I am not flying I can tuck everything away. Plus, the board adds enough weight to keep the controllers from sliding all over.
Hope something in this response was helpful. Good luck!