You don’t need to worry about formatting. The installer can do that for you as part of the process. Just make sure you merge all of the existing Windows partitions into one, then let the installer partition as needed. I’m not 100% sure about the Steam question, but I think I remember reading somewhere that this is the case.
Keep in mind, Linux isn’t magic. If the hardware sucks, the hardware sucks. Games can take a lot of resources. Just make sure your old LAN party laptop can handle the games you want to play on it.
Side note: Mint Cinnamon is a perfect choice for a starter distro.