I have an app that I want to call "Donald Trump". Now that Donald Trump is a political figure, is it okay to name my app after him (first amendment)? Or does he have a trademark or something over his name? I don't want to get sued but I'm wondering if his political status allows me to use his name openly.
This question seems to be requesting specific legal advice and therefore is not appropriate for this website.
I can however answer two generic questions, so long as you understand that I am not your lawyer, that you should consult a lawyer before taking any particular action, and that the answers I am providing are generalized discussions of the broad topic you have introduced intended to be merely academic in nature.
Trademark law generally is intended to protect the holder of a mark from consumers being confused as to whether the mark owner has produced or endorsed the product. The answer to any trademark question will turn on specific precedents of the sort of use, but that concept should serve as a first principle for guessing the likely answer.
The fact that someone is a public figure provides them with fewer rights per slander and libel laws, but probably doesn't matter very much when it comes to their right to claim their own intellectual property.
Would you ask, does Al Gore have a right to enforce his patent on the internet (jk), given that he is a public figure? No that's preposterous. So why do you ask about someone's right to protect their other IP? The one "exception" to this that readily comes to mind is that copyright law affords a "fair use" defense which treats political satire more permissively than some other uses; but I don't know believe that has anything to do with trademark law.
Summary: I am not your lawyer and Don't confuse the torts of slander (which directly relate to speech) with the property rights afforded by IP (which only incidentally relate to speech).
Note: the comments above regarding ability to finance suit, strike me as counterproductive. That's is always an issue, and the generic question (i.e. the only appropriate questions for this site), shouldn't implicate such a consideration.