Results of yesterday testing:
Removing battery didn't help. I didn't have another power supply with 8-pin CPU connector, but when I plugged just 24-pin connector from another PS LEDs were normal, I could boot, but obviously CPU fan was not spinning.
I tried the same with Chieftec, and I was also able to boot. Plugging 8-pin CPU connector was causing LEDs to blink fast, but after few disconnects I managed to get everything working and booted normally. That is, until I shut down my PC 6 hours later, when same blinking occured.
Obviously problem is with that 8-pin connector, either on power supply, or maybe power unit on the board is dying.