I have a laptop with a more feature-rich version of the same motherboard, that small green board in the bottom left is most likely your storage. It is connected to what I believe is a proprietary sata connector. You can double check by looking for the header marked with the code jhdd. If I’m correct, you can probably buy an adapter for your model of laptop/motherboard, and connect that to your new SSD.

A computer has limits built into it so that it never gets hot enough to outright destroy itself (although it can wear out faster). You’ll be fine, you won’t break it by playing games on it. As other people have said, it would be a good idea to invest in a cooler, if only to make your gaming experience better.