>>>would self reboot sporadically for few months >>>

I assume you mean that it spontaneously rebooted while you were using it. Have you tried to toggle the screen on with the Fn+F6 or Fn+F5 toggle screen combos?  Jack E/NJ

>>>3. Taken out HDDs>>>

Two more suggestions, Kaq.

First, when you removed the HDD, did you find any dust bunnies inside? If none, did you try to vaccum or blow them out anyway in case they were hidden? If no, I suggest that you vacuum the air intake port and use the vacuum blower on the air exhaust port.

Second, to check your video adapter, did you try to HDMI or VGA out to an external TV set? You might have to turn the TV set on with the Fn+F5 toggle key combo.

Jack E/NJ

>>>The TV displays No Signal Message.>>>

Do you hear anything that sounds like a "beep" code immediately after trying to cold boot? Jack E/NJ

>>>Only other sign (good or bad) is that I have to keep pressing power button for three to four seconds to turn it off.>>>

Pressing & holding the power button for 4-5 seconds is the normal way to force a shut-off. Accordingly, it's a good sign at least for the power button operation.

Are there any audio sounds like welcoming tones/melodies to suggest that windows is actually booting even though there's no display? You may have to wait several minutes to hear any welcoming tones because you've forced a windows shutoff which automatically delays the boot routine if there's no keyboard input.

Jack E/NJ

OK. The graphics card partly depends on shared PC memory. After removing battery, I suggest you first try to re-seat the memory modules a half-dozen or so times to clean the contacts. They should be accessible behind an L-shaped panel on the bottom held with screws. Jack E/NJ  

Kaq>>>I guess even bios is not loading.>>>

Yes, the bios is suspect with no beep error codes when HDD & display errors are obvious. Either the bios chip has failed. Or its firmware corrupted. If the latter, a bios reset to default status might be possible.

When you opened the memory module compartment, did you see any jumper pins around the peripheries of the memory module slots.There might be 3 pins in a row, 2 of which may already be jumpered. These might even be labelled 'bios reset' on the mainboard.

Jack E/NJ  

OK. Next, power down. Unplug charger. Remove battery. Press & hold power button down for several minutes to try to drain any residual charge that may be causing at least an apparent POST failure. For good measure, you might want to repeat this process a few times over a period of several hours before replacing the battery and testing the system again. Jack E/NJ

Not a good sign for the BIOS. Another thing we can try to wake it up --- power down, remove the battery. Remove the memory modules. Then replace the battery. Press and hold down a bunch of random keys on the keyboard. Then power up to see if you get anything that sounds like a BIOS keyboard & memory beep error codes. If still no signs of life, the BIOS chip itself may no longer be functional. Jack E/NJ

If it were mine, I'd definitely try to flash the BIOS. You really have nothing to lose at this point. Do you know how to make a USB boot stick upon which to copy the flashing files? And the flash Fn+Esc process? Jack E/NJ

OK. First we're gonna try to make the USB stick bootable. Place the stick in a USB port.  Then open the cmd.exe prompt as an administrator. At the prompt, enter diskpart. Then enter the following commands at the diskpart> prompt.

list disk
select disk # (# is the number listed for your USB stick, might be 2 or 3)

list disk (make sure the correct disk, the USB stick, is selected with an asterisk *) 
clean
create partition primary
select partition 1
active
format fs=fat quick
assign
exit 

Now copy the four files you found in the DOS folder into the root directory of the USB sttick. Then put the stick into a USB port of the turned-off 8735G. Press and hold the Fn+Esc key. Then press the power button while still holding the Fn+Esc keys. After about 20 seconds, release the Fn+Esc keys and hope for the best. Report back with results.

Jack E/NJ