It really, really, helps if you have some other kind of skill. Learn to program or make art as well as scripting. You're just more employable if you can do something else - and it also helps you fit into positions which involve a bit of both worlds.
For someone with an artistic background, level design is a pretty straight foward path. Work as a level designer at a studio for a few years and try to work your way up, if you still want to "design" games at that point. You don't need to make your own game from scratch to get into design. What you do need to do though, is…
Yes and no. While there are such positions, they are few and far between. The only people who ever really get to "just design" day in and day out are industry legends. And most of those designers have spent quite a bit of time in the trenches producing levels or writing code. Yes, this would make a lot of sense. Paradan…