As you already know, it's not the monitor. By turning up the contrast and brightness to full, you see a pattern on the screen that fills the screen, and with re-trace lines (diagonal lines). That is normal for these machines and indicates that the basic monitor (TV) circuit is working. But it could be the monitor's video input circuit, or maybe the video driver on the logic board, right? Or, if the video circuit wasn't providing video, HSYNC and VSYNC, would you get the retrace lines?
A bad ROM isn't the first thing I would look at either. If you had a spare ROM, swapping it would be a quick test. But, lacking such a spare, there are better things to test first.
Based on your description, the CPU is working, at least initially. Since the floppy disk light comes on, it looks to me like the CPU is searching for a floppy. That light doesn't come on unless the CPU asks it to. The disk lights may flash on power up, but then a moment later the CPU looks for a floppy, turning on the LED for a few seconds. So in the basic sense, the logic board is working.
PM me and I can send you the service manual for the Model III. The manual has schematics and a pretty complete theory of operation.
In the mean time some basic tests to try. Raymond's tests above are best if you just want to probe the logic board, or try the following.
Did you get a CASS? prompt like described before? I expect not, but, if you did, then you know something is wrong with the disk sub-system causing the computer to lock up.
Do you have a working floppy disk to boot from? If so, put it in, and see if the floppy drive does more than just turn on it's LED and then go off. You should hear a few head movements. Also, I think TRSDOS 1.3 (maybe others) does a quick scan of the other floppy drive, causing the LED on the top floppy to come on.
If you don't have a working floppy disk, maybe you can do something from BASIC (boot into CASSETTE BASIC) that will show signs of life. Try making the cassette on/off lead transition with a CLOAD command.
To get into cassette basic, hold the BREAK key while pressing reset. Press Enter twice (well, 3 or 4 times just to make sure), and type CLOAD <ENTER>. If you have a cassette drive hooked up, with the play button pressed, CLOAD should cause the cassette drive to start playing. If you don't have a cassette, then probe the cassette REMOTE line from the 5 pin DIN connector to see if the line transitions. Info on the cassette cable can be found here:
http://www.8bit-micro.com/data-cable.htm
Alternatively, if you have a printer (that you know works), hook it up, do the same above to get into BASIC, and type LPRINT "Hello" <ENTER>. Hello should print on the printer.
I can't think of other BASIC commands that would show sign of life, so the above may be your best options.
Being a TV guy, you might want to check the VIDEO, VSYNC, and HSYNC. They are on connector J5. Video is pin 2 on J5 (or pin 13 of U52, the video generator), HSYNC is pin 6 on J5 (or pin 13 of U23), and VSYNC is pin 4 no J5 (or pin 12 of U23). U23 is a 74LS221, with 8uS given as the HSYNC value, and 693uS given for the VSYNC value (based on the RC circuits on the 74LS221 I assume).
Beyond that, it boils down to the kind of tests Raymond suggests above. Raymond's tests might be easier than my suggestions being that the machine is already disassembled I assume.
I hope this helps.
Mike