Have you tried clearing the CMOS memory correctly by using the clear CMOS link on the board with the power cord DISCONECTED?
Did you try pjmelect's instructions?
Socket 1155 doesn't support intel core 2 duo.
So if you are gonna switch to 1155, it'll be easier to do a full new system, since 1155-based motherboards don't support DDR2 RAM either.
But yeah, that's really last resort.
Make sure that it's not the graphics card failing (cos that is extremely cheap to replace in comparison).
If it does end up being the motherboard, you will have to make a decision to either buy a new socket 775 motherboard (which is also very cheap as well but like I said before, lacks upgradeability) or buy parts for an entirely new system.