A lot of what woot does falls under fair use or parody. An easy way to think of it is
that if what you make is could ever be confused with the original, then you could land yourself in legal trouble. For instance, Selling a this would get you in trouble,
whereas this is obviously a parody
That doesn’t mean Disney won’t send you a Cease and Desist that threatens legal action though. They have a lawyer send you a scary looking letter saying “we will sue you”, even if they don’t have a legal case. They have billions of dollars and thousands of lawyers- they can afford a long fight, and you can’t. So most people back off
Woot will often turn down designs that they don’t feel comfortable with, and you’ll sometimes see shirts disappear from their catalogue, like all of their LOTR designs.
But they also have a legal team, so they can push the envelope more than you or I could.
Another thing to know is that copyright policing is pretty much up to the copyright holder, and some are more lenient than others. Woot has a general idea about who not to mess with. Newline Cinema (LOTR) might be d**ks whereas BBC seems to be pretty safe.
What I do is stay away from copyrighted words, phrases, names, and logos. I would never put “Dr Who” or “Gallifrey” on a shirt, but it’s harder to copyright a man in a tweed jacket, red bowtie and a fez. The Police Call Box existed before Dr Who, so that too could be used. You can also change, or obscure copyrightable portions of a character, like the bat logo on Batmans chest to cover your bases. Woot will print a caricature of Captain Picard, but not the Star Trek insignia.
There are lots of ways to get around it but it is always kind of a risk.