Westinghouse 20" LCD TV

Do the following, and see what happens:

Switch everything off (Mini and LCD)

Make sure the connectors are tight (Mini-DVI-VGA-Cable-DVI-LCD)

Start-up the mini (leaving the LCD off!)

Once you are sure the mini has started, switch on the LCD. Without a monitor connected, the mini SHOULD start-up rendering at 800 x 600 @ 60 (32 deep).

You should have a pretty small, pretty bland but pretty real OS X rendition on your LCD. Now you can start trying to get the resolution you desire. Your LCD monitor would most likely have a list of the resolutions it supports. Try the one you want via System Preferences -> Displays.

So many people post similar problems. I have recommended CSCREEN because its free BUT the problem is that it requires a little (very limited) skill in the terminal.
Others use DisplayConfigX or SwitchResX, both sharewares costing $10. All three can be found at the usual places (versiontracker.com and macupdate.com). Searching these forums, you will find posts by myself describing how to use CScreen.

Dunno - others may be of service here. I just used cscreen and a little applescript when I needed a wide aspect ratio or strange display setting.