motherboard - Desktop will not boot, not even to the BIOS

My desktop, assembled back in December 2011, specs are:

Gigabyte H61M-S2P-B3 rev1.0

Intel i7 2600 3.4GHz

Kingston DDR3 4GB 1333MHz

Seagate Barracuda 500GB 7200RPM

Windows 10 x64 Pro updated to latest service pack and other updates

iBall JSW 250W PSU with the i7272 cabinet

2.1 Stereo on VIA 1708S

Dell S2240L 21.4" Full HD display

The problem:
Over the past week the computer had gone through these stages.

1.Some lag and frame drops while playing videos then audio video hangs and PC crashes then,

2.I disconnected the HDD and restarted but the computer boots to bios and crashes within seconds, restarting again crashes then,

3.After a few attempts when I start it again there is no display and PC is stuck in boot loop i.e. CPU fan starts-no display-crashes and again CPU starts-no display-crashes.

I have tried cleaning the RAM and reinserting but step-3 occurs, tried resetting bios no effect again step-3

Oddly enough, when I disconnect the power and leave it idle for a couple of days then the problem starts with step-2. Also, I don't know how but back panel 2 USB ports out of 4 stopped functioning despite limited usage, same with the 2 front panel ports.

What could be the problem and what could be the failing component?

Kindly guide me to pinpoint the problem.

Also if the motherboard is failing I'd preferably go with H77 or Z77 chipset instead of H61 as an upgrade, so will it be a good idea to do so?

Thank you.

Update 1.0: After trying the method mentioned by konstantin, to me the problem seems to be with the BIOS and I'm able to sustain the boot for over 1 minute, but can't be sure, as seemingly there is no damage or burns to any component whatsoever.
Lastly, is it advisable to try the Q flash utility for reflashing the BIOS?

Supplementary question: What generally happens if one powers up the system with ONLY power supply, CPU and fans in place, no ram no HDD in place, only CPU and PSU?

In my case what happens is that the CPU powers up for about 10 seconds with no display on monitor and then shuts down and then again restarts as in step3 above. Also the same thing is occurring with the RAM in place. Any ideas or advice?