I'm very new to the world of screen writing, so I was wondering if most script writing jobs are free-lance, or do you work for a specific company (say, Disney), and have a paid position with steady work? Or do some people do both? I would prefer to work for a company since I do will with structure -- how exactly do most script-writing jobs operate?

If you can find ANY job at all as a paid screenwriter, you're part of a very small percentage of writers. Sure, there are some writers here at SS, for example, who have got paid for their writing, but it's not the sums anyone could live off. Writing for a studio? That's probably every screenwriters' dream. Very hard to pull off.