Try setting your Aliasing and Anti-Aliasing setting to 1 or off, sounds like a problem ATI cards had back in Beta and early live. Turned out to be an ATI Driver issue. Can also try rolling back your video drivers. Thos with Nvidia cards can tell you to which driver set to roll back to.
You may also want to force a scan of all game files; open game to login screen, click Options and select "Force Verify."
DOH, just re-read your post. sounds like it is your system. Since you did a re-install with Win7 did you re-install DX 9c latest version? STO is DX 9c not DX 10 or DX 11. Yes DX9c, DX10 and DX11 can co-exist under Win7.