Whenever I copy/cut and past something, or fill it with a colour (ie white), there's this hideous gray outline around it.
What's up with that and how do I fix it?
It's probably the way that you're selecting things. The gray line you see are artifacts that are caused by anti aliasing. Read m0g's tutorial on Vector Masking and then you should be able to sort of your problem. You can find it here.