Do web designers steal code?
No.
Have web designers stolen code?
Yes.
Some designs can be taken by doing a save-as, but that only gets a static page. More and more sites are becoming dynamically generated, but I suppose the design aspect can be taken though there's still issues with it. Depending on geographical location a person could file a lawsuit against someone for stealing their design if they can prove it's theirs.
Real web designers only get
inspired from other designs. Some designers offer their designs (for free or paid) as a template that people can use, so wouldn't be stealing in those cases. To be honest, it's almost as hard to adequately steal someone's design as it is to come up with one. I remember someone posting at Mac Rumors who had their design stolen by a guy who didn't even take the time to remove the original designers name from the code. It even copied a contact form that didn't work in the least. It was very pitiful.
(
The below paragraph is more general statement for designers, not to the OP as he's no the designer)
If you're thinking of stealing someone's design, just stop thinking. It's a waste of time, you'll look stupid, and there's so many free sources out on the web you'd be plain stupid for ignoring them and putting yourself at needless risk of a lawsuit. If the person you steal from is also a hacker they may come back at you with a vengeance that'll keep you from sitting down for a week.

Not that I've ever done that.