I just installed Quake 2 on my computer with Windows 7 Pro 64 bit
I changed the compatibility to Windows XP SP3.
When I load the game, I do not get mouse control in-game.
The mouse still functions but outside the game.
I can only use left, right, up, and down with keyboard.
Also, I do not have full screen, I still have desktop taskbar visible.
I loaded the update q2-3.20-x86-full-ctf, but still have the same problem.
You should try one of the newer Quake2 clients. On Windows, I would recommend R1Q2. That's about all I can offer; I provide support for Quake2 on Mac and Linux here :)