KingPookybuttz: How do I stop gog.com from starting up when I first start up my laptop
This is a Linux thread and GOG Galaxy hasn't been released for Linux but, if you meant the GOG.com website, then it would be something misbehaving with your browser's session-saving and you should try navigating the tab to somewhere else.
KingPookybuttz: and how do I delete games out of or off of my library?
If, by "my library", you mean the listing of games you own, you can only choose to hide things. Like Steam, they assume you paid real money for things and don't let you permanently revoke your own access to what you paid for.
If you actually mean "the set of games that Galaxy has installed on my PC", I can't help you there because I've never used Galaxy.