PC randomly doesn't boot after sleep + no POST (no BSOD)

Ok well a simple solution would be to fit a new fan on the cooler but before that try turning the fan with your finger (machine powered off) if it is stiff or it has some sideways movement then I would be reckoning the bearings are out of lubrication or are shot.

Now if it is lubrication then I personally have relubricated the bearings in many fans before and to do that you remove the fan carefully peel back the label on the back of the fan and it will reveal the bearing usually a porous bronze sleeve which can then be dabbed with a lubricant of some sort using a cotton bud. The lubricant itself I have used WD40 and even light sewing machine oil in the past - but you only need a minimal amount applied to the bearing and then restick the label back. Just put the lubricant i a small container and dip the cotton bud into this this ensures that the lubricant goes directly to the bearing.

Optimally it is best done with the fan spinning - in my computer room I have an old PSU set up to plug components to ie fans in cases like this to do this but you could try just doing it by hand.

If the fan is out of whack ie has some movement sideways (wobbles even slightly) on it's bearing then just replace it - cheap to do and as you know they all come in stock sizes just a matter of measuring the one you have attached to the cooler.

Just out of interest what are the temps when the machine running? us Speccy Speccy - Free Download - Piriform

No, no, no: the RPMs are perfect when I boot the pc normally, but as soon as I put it too sleep next time they're halved. Restart the pc, good again, put to sleep, halved again. The RPM curve is the same, only instead of going from 0 to ~1500 RPMs it goes from 0 to ~600. And the numbers aren't just skewed, either: the temp does climb 10-15°C and the CPU complains after a while. The fan itself works fine, but the controller just decides to not work properly after sleep.
It seems this is not an uncommon issue, too. As for what I'm seeing, it seems it's a mobo issue, and we can't do much about it, although I would really love try to see if there anything at all that could be done... like, who knows, maybe quickly rebooting part of the faulty bios after sleep.

Ok I thought you meant it ran slowly in any case - I misread that and in that case then yes it is either CPU or a mobo sensor issue.

Now in my mind on a cold boot the CPU would necessarily being involved in quite a few processes like loading the RAM and system in general. It then occurs to me that on waking from sleep because the machine is already loaded as it were then it would follow that the machine is already at some level of operating temp and just doesn't need that extra speed for the CPU cooler fan.

Having said that when you bring it out of sleep and then give a bit of heavy duty use say for example a full AV scan does the fan then speed up because the scan would be making the CPU work harder. I know my machines do this - although I rarely have sleeping but do know that again starting a scan of some sort instantly speeds up the cooler fan especially in my laptops.

Just an afterthought the fan plug how well does that fit onto the pins on the board? If you just hold it and try move it from side to side back to front is there any slop noticeable - could be that heat from the mobo while sleep is on is letting connectors expand away from the pins on the board. If the connectors inside the plug casing look to be a little large for the pins then I would try making them a little tighter by gently easing the sides together - it doesn't have to be much but enough for the connectors to make good contact with the board pins. Again even those pins do they look straight or bent and the joint to the board any signs of scorching or being a dry joint - if you are not familiar with that term it means the soldered joint has come apart and can lose all contact or have partial contact with the board printed circuitry.

I am only surmising some of this but it is something I would check myself on my machines and because that particular fan connection is a four pin connection as opposed to the ordinary case cooling fans which areas you know three pin connectors as there is control connection .

Ok mate now please be VERY careful with beta BIOS updates although having said that I see that board has a dual BIOS capability. Just one thing puzzles me though and that update you mention (F4a) seems very different to the one I can find on the Gigabyte site?? GIGABYTE - Motherboard - Socket AM3+ - GA-990FXA-UD3 (rev. 1.0) obviously you will need the European but it might be an idea to check in one of thsoe Gigabyte forums on this link first

Now you can see from this https://en.wikipedia.org/wiki/Computer_fan_control the issue is the Pulse Width Modulating circuitry (PWM) which is why the cooler fan has that extra connector so I have two theories on it and that is the BIOS is not right and is not connecting with that circuitry or the circuitry once in essence shut down to a slow speed is not switching back to it's correct operation once the sleep period is stopped.

From my experience with Gigabyte - my old Sandy Bridge build with their brand board - was that their support was ordinary at best and plain hopeless at worst.