If still the same, the motherboard I/O chip might be busted by a power surge/ESD, but if I remember right you can also take out the CPU and re-seat it. Those AM1 CPU's are a little flaky, they have everything on them including the south-bridge, and just a bit of oxidation on the pins in a more humid climate can cause unforseen problems. Its worth a try, if nothing suggested in this thread works, your last bet unfortunately is RMA/warranty

I have seen similar issues caused by a standoff in the wrong place causing shorts. You have taken the board out of the case but if something under the board shorted it may have created a carbon trace or bent over one of the capacitor/component wires soldered through the PCB. Give it a good going over, I would start with the underside of the board nearest the 24pin connector.

Good luck

the motherboard should trigger/short pin 16 only when a wake up event happens (either via button switch, either some wake on X event). In my case, pin 16 is constantly shorted, sending power on signal to the psu all the time, which is wrong. the pc works "fine", it just cannot properly power down to "off" state (sleep, hibernate or whatever), since psu is constantly in full awake state. 

You've said after shutting down your system... the psu fan and cpu fan is still spinning? this is caused by bad HDD. 

Now you have different issues which " When I plug in power cord, only the psu powers up (fan starts spinning), but power usage meter is showing 30W draw in this state. I tried this: if I unplug the 24pin cable from motherboard, connecting the powercord does nothing. Similarly, toggling psu power switch does nothing. As soon as I connect 24pin the psu powers up, clearly indicating that the motherboard is giving "orders"... ???

Your first post and your reply to me has two different issues.

Can you post a video here in shutting down your system?

If I connect multimeter to marked pin and left/right neighbor I get a beep 

this is where my electronic knowledge ends... is this fixable?

Its usually pin 15 (ground) or another ground pin/plane (I dont know exacly) and Pin 16 (PS-ON) that start the PSU, but the motherboard circuitry receives power even when the PSU is "off", that is the standby power, true power off is achieved only when the PSU is manually switched off, until then the motherboard receives power. 

Can you try checking with the multimeter and the motherboard out of the case with all cables disconnected if it still reports continuity between any ground pin and pin 16 from the ATX connector? if yes, then there is a problem with the mainboard, because no continuity should be present if pin 16 is not shorted to ground, and that cannot happen if the power button on the mainboard is not shorted, before doing these measurements also please disconnect UEFI battery 

ffs

When I plug in power cord, only the psu powers up (fan starts spinning), but power usage meter is showing 30W draw in this state. I tried this: if I unplug the 24pin cable from motherboard, connecting the powercord does nothing. Similarly, toggling psu power switch does nothing. As soon as I connect 24pin the psu powers up, clearly indicating that the motherboard is giving "orders"...

The psu is brand new bequiet sfx power 2. Before that, I had this setup in a different itx case with integrated psu and the behavior was the same.

So, I'm pretty much blaming motherboard...