My motherboard is working, at least for now.
When it failed to boot I brought the entire PC to some random local service. Next day it was "fixed" = they said that all they did was to reload the BIOS again. I remember that I read somewhere the procedure and will search for it, may be helpful for others. I personally thought that the problem is hardware only and didn't bother with BIOS reload. Anyway, when I got it home, the motherboard had the same symptoms - when I press the power button the fans were spinning for a while and then turned off. Brought back the PC and told them to search why the BIOS is crashing, is it something in the BIOS chip or some element on the motherboard. Actually I'm not sure if this is the case because it may be just something preventing the motherboard to start. The second time when I went to took my PC, I asked them a few times what was the problem and the said that they don't know and suggested "try not to tweak it too much". But that's not the case. They didn't told me what exactly were doing but I suppose that they just reloaded the BIOS again.
Maybe they didn't want to bother too much in order to reduce the cost for their service because it's old motherboard for them it was not worth to repair it - indeed they charged me just 15 euro in total, for both of the services. I know that my motherboard is extremely picky, but suppose that the problem will appear again and will look for another service just to make sure that it's not hardware related. Meanwhile I checked every thread for Asus M2N32 on elektroda.pl - the forum mentioned by gayLaser. To sum up - the problems may be in capacitors: PSU; you may need to clean contact points for the BIOS battery: in another case there were some incompatibility between RAM modules and the motherboard itself but the beeps were for GPU failure; and CPU power supply MOSFETs and their controllers.
I still can't run AMD GPU on the first PCI Express slot. Tried changing the BIOS settings for which slot is primary but no luck at all.
When I insert some old Nvidia GT something (without additional power socket) the board POST successfully. Tried Quadro K6000 in the first slot - successful POST when the PSU cables are inserted into the GPU and when they are not I got a message that this GPU needs additional power supply. So I think that the slot itself is OK. But I want to run Firepro V7900 in the first slot - when I try it, the GPU fans are spinning at max rpm and I can hear one long and three short beeps. Tested it with three brands of RAM memory just in case if it's like the case on elektroda.pl but no luck. But the GPU is working fine on another motherboard and also when it's placed on the second PCI Express slot the motherboard is able to post.
The same case was a few years ago when I was trying to run R9 280x on this motherboard - it was working fine on the first slot, , but whenever I try to use it in the first slot, the motherboard refuses to POST. Back then I gave the board to a friend of mine which replaced almost every capacitor but no change in the symptoms. For some reason the Nvidia chipset doesn't like AMD GPUs
By the way, does anyone knows the beep codes for these motherboards - mine is M2N32WS Pro.