Sent the photos off to you!
I've had a good look at those close ups, everything looks good, no obvious signs of damaged resistors, chips or capacitors. The ROM looks like it has a patch in the bottom left of it where moisture has been on it but I doubt that has anything to do with the problem. You could try taking the ROM out and putting it back in just to make sure but typically you would get the red screen error if anything was wrong with the ROM.
The Agnus is an 8372A, which is the 1Mb chip RAM Agnus so you must have a rev 6a / 7 board probably, unless someone has replaced the Agnus before - but from looking at where pin 1 is I suspect it is an original chip on rev 6a / 7 board. It doesnt look pushed all the way in to me, I wonder if someones done the double sided sticky pad job on it. I recently bought an A500 where someone had removed the agnus, damaged the socket and then 'stuck' it back in the socket with a double sided pad, which is fine for a period of time but it means the chip doesnt sit as deep into the socket as it should. CSYNC, crashes and floppy drive errors can be all caused by the Agnus not sitting correctly (or having an internal fault).
I've rechecked regards -12v and the minus 12v shouldn't affect anything other than comms (serial port and parallel port) so I wouldnt worry about that for now.
Is it a UK console that you've had for years, not a US version or something?
You can typically pick up an Agnus 8372A for between £5 and £10 on EBAY, but I don't see any UK sellers at the moment. The difficulty with the Agnus is removing it very very very carefully in such a way that you don't damage the socket. Ideally you should use a PLCC extractor tool but they aren't always easy to use. The alternative and dangerous (if you dont be careful and dont know what you are doing) method is to use a jewlers screwdriver (very very thin screwdriver and put it under the little hole (looking at your photo) in the top right and lever under the chip a little (and I mean a little) bit, like a quarter of a millimeter of movement, then do the same with the bottom right hole / gap / slot call it what you will, again levering just a tiny tiny amount, repeat that process 4 or 5 times for each of the 2 corners and provided you hardly use any force at all it will lift out.
If the socket has become brittle (which they do over time) and you press too hard you will break the socket, which means desoldering the 84 pin PLCC socket and putting a new one on. I've done this recently for the 500 I bought - it was in a shocking state.
I think you should try another TV to see what if anything is different on the video outputs, check the pins of the video sockets on the back of the amiga as well to make sure none are bent and shorting on each other etc - i've seen that before from people ramming in the modulator when its not straight etc. If there's no difference on another TV, and pressing firmly on Agnus to double check its totally making contact (you could start the A500 up whilst you are pressing down firmly on it with your thumb), and you've not got another power supply to try (to 100% rule it out), then you are looking at a faulty chip or other component, so its then a question of how you go about getting replacement bits of EBAY etc, or just get a working / semi working 500 off EBAY as a doner.
Do you live in the UK or elsewhere?
EDIT: Removing Agnus - on your photo the slots to get under the chip are TOP LEFT and BOTTOM RIGHT, not TOP RIGHT!!
