It could be two faults.
If the computer doesn't boot when powered by the electricity mains, that would seem to point to a power supply problem. It can't be heat-related if it does this even when the computer is cold. The games freezing could be related to the power fault too. Maybe the battery going down a bit after that time, or not charging properly?
If the games fault is due to overheating, maybe something
like this will help. However, if they have been running properly in the past, then that would also seem to point to a fault, maybe the power supply or overheating. The air-duster might not be hitting the right spot(s). There could perhaps be dust on the processor or fans that the blast of air isn't reaching. The processor might ordinarily cope with the heavy demand some games put on it but possibly won't if it overheats due to dust.
On the other hand, if the games have always done this, then it could be that the computer isn't up to running them as has been suggested.
In any event, I think the bottom line is that the computer needs looking at by someone local to you who is experienced in sorting out computer faults. It's often difficult to be right about computer problems without actually being there. I'd suggest a trip to a repair shop.