Hot exhaust and warm case areas are normal depending on CPU usage. A hot exhaust means the CPU is NOT hot --- it's being properly cooled. The heat generated by the CPU is being properly carried away to the exhaust port by its fan like a car radiator fan carries excessive heat away from car's engine. Furthermore, the CPU will throttle or power off if its temperature reaches excessive levels. In these cases where heat is disabling the machine, rather than disassembling it's always best to first try vacuuming the air intake ports and blowing through the exhaust ports a few times to dislodge dust bunnies or other debris that might've accumulated inside.

As for the issue you have now, I suspect an electrical short or similar current leak due to a misaligned screw, pinched ribbon cable or loose connector is tripping a mainboard's circuit breaker. You should immediately stop trying to force the system to turn on --- the circuit breaker may eventually fail and fry the mainboard or a critical component. You need to disassemble the machine again and carefully re-assemble making sure every connection, including the CPU pins, and are properly seated. Then before reassembling the entire case, put the machine on an insulated surface and test if it will stay on.

Jack E/NJ