Pull your ram out and clean the contacts. Make sure they are completely seated when you reinstall them. On the bios screen find and select the default settings.
I had a similar rebooting, shutdown, weird screen problem with a Dell and it turned out that the RAM was not fully engaged in the socket. The steamy hot CPU is a different matter. That's why I mention a default bios setting.
will be interesting to see what $$ they want for a mother board compared to a new laptop
you may be better off with a new laptop and get a decent warrenty.
Look for something a bit better quality than a Gateway. Definately wouldnt be my first choice in laptops ( or any form of their brand of puter)
after working on them for some years.
If you have a bad cap i would remove the parts that look faulty or test faulty via the bus. Get replacements and re solder. U can test caps and resistors via the bus so get a multimeter and start there.