Monitor doesn't display anyting

Sounds like the display not an internal hardware problem came up on you there. If you still see the standby indicator light staying on standby then you know the monitor let go. If the lighted power or other led remains amber if one is present and that doesn't change at all during the initial post tests then suspect that went.

You won't hear any beep code for that unfortunately. How old is the monitor itself. If it's been in use for a few years already I would check to see if that is the actual problem by trying another if you have one you can at least borrow for a quick test to see if everything else is running normal.

Video cards can also fail without an audio alert at times. Rechecking the connection for the additional power feed and getting inside with a can of air cleaner to blast the PCIe slot out good for a reseat of the card might be simply a card lifting up in the slot problem.

When no signal is being displayed like you would expect inbetween post screens for a few seconds that's when it should be amber. Once a normal post and startup screens are active you should be seeing it change to green if the card is sending out the normal signal.

The same applies for the 20 minute turn off display setting seen in the power options. When no activity is seen after the default 20min. the standby mode kicks in when Windows turns off it's own display.

The analog input working is just that another input known to be working. Once you see the vga or dvi input switching off or not coming on when a normal signal is present the switching mechanism in the monitor maybe letting go. I saw that on one monitor a few years where you had to startup a few times just to get the worn lcd to kick on. That's when I knew it was time for a replacement.

I've recorded when it's happening, and it's only whenever I play intense (aka non-indie) games on the PC. I can get a couple of minutes in before the screen goes off. If I alt-tab out, the screen comes back on straight away with the game crashed.

is my GPU overheating in this hot UK heat?

Have you tried monitoring the gpu temps with anything like GPU-Z or PC Wizard 2010? CPUID - System & hardware benchmark, monitoring, reporting

The PC Wizard program shows both cpu, cpu core temps along with those for your video card and usually right on with what is seen in the bios hardware monitor with a quick restart. A card overheating especially if things are turned up a bit would see black screens even hard crashes at times while gaming.

Try seeing what the temps are with that one to see if "hot" is what you are but don't want to see.

I downloaded the program you linked and it says my Intel Core 2 Duo is at 29 degrees C and my Nvidia 7900 GS is at 72 degrees C. It says the fan is at 20%.

Those times are on rest, not playing games.

TF2 gets 90 degrees C.

Are these good/bad temps?

edit - I've opened the case and have got a fan pointing at the inside. Seems to have stopped the black screens, but this isn't a permanent solution!

At 90C for a card temp that is a little high while not being disasterous! The cpu temp is excellent by the way where you can rule that out but now have to bring the video card's temps down.

The MSI Radeon 5750 1tb model runs about 62C when running some intensive video and graphics here while not seeing any large drop after suggesting a little extra cooling here wouldn't hurt while it still is remaining stable. But at least you have the problem isolated and know what to do for it.

The 2010 version of PC Wizard is certainly a good one to have. That shows the main cpu temp as well as what is seen for each core even on the quad here and seems to remain up close to what the hardware monitor shows.

Welcome try downloading Evga precision. This will allow you to change your fan speed, might want to run it at 75 to 100% while playing to see if it helps lower the temps and stops the problem this will give you a good indication if it is heat related. Fabe EVGA Precision 1.8.1 download from Guru3D.com