How can I keep these hook mounting screws tight?
The best thing you can use is threadlocker, which is a special glue designed to keep screws from coming out. A drop on the threads keeps them from slipping. There are different grades and colors, you'd probably want blue.
If you don't want to buy threadlocker you can try a drop of superglue. One of the comments say you can try nail polish, I haven't had much success with that trick but if you have it on hand you could give it a try.