Let's say I wanted to make a T-shirt with a caracature of Bea Arthur and sell the T-shirt, with her name on it. Does this violate any trademark law by essentially using her likeness for something I would want to make money from, without a representative from her estate granting permission?
Is there a difference if the person is living or dead? I know with music and literature you can record and sell music/books from the so-called public domain. Does this apply with old, dead people, like George Washington?
Could this classify as a parody and have a better chance of being protected?
Celebrities don't trademark their images, but what does protect them, even the dead ones, is state laws governing celebrities' "right of publicity," which is their exclusive right to profit from their own images, signatures, voices, etc.
Those publicity rights aren't absolute, however, they give way to news reports and sports reports and 1st amendment rights of expression for true artistic creations that don't simply capitalize on someone's fame. A caricature/parody that transforms the image, and comments on it in some way, may fit that description.
See these other similar questions and answers, and then see an IP lawyer. Bear in mind that with deep-pocketed celebrities or their estates, they can easily legally stomp all over a non-rich entrepreneur.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.