Hi there :D
1: what game engine would I use?
Well, you should decide what kind of game you are going to create. Is it 3D or 2D? choose proper engine which could be used to create your choice.
2: I have a Intel desktop but it's an i5 with a integrated graphic chip, no actual graphics card. Can I still make a game with that?
I'm not familiar with specs like this but there should be engines for low spec PCs.
3: if I'm new to coding and I don't plan on having a game done for at least 2 more years, what language would I even learn first?
This should be fitted to your chosen engine.
4: I'm not the best at art or drawing but I'm always practicing to improve. What applications could I use for making my own assets? I know they are gonna look a little crusty. But if I can practice now it will help later!
Again, as mentioned in #1, you need to decide if your game is going to be 3D or 2D. From the answer you can choose proper apps to create your assets.
5: how do you stay motivated?....
1. Create game that you really love. Don't make game because it's popular, famous or current trend! Make a game that you really love even if only a few people play this kind of game.
2. Have enough break and sleep.
3. Play some games preferrable ones with same genre as the one you're making.
Good luck with your game :D.