Don;t know about bedbugs there, since I've never stayed there..but, I'm guessing it would have been listed in recent newspaper reports or in the Trip Advisor hotel reviews pages.
I doubt you get anything back for your troubles.
As QB suggested, read the reviews posted here on TA re: The New York Palace Hotel. If there is a bedbug problem, the reviewers will say so. Although sometimes people who encounter them don't know what bit them and will often write that they were bitten by fleas--so watch for such language. Also, recently there was a thread in this forum entitled "bedbug hysteria?". Look for it--it has helpful info on how to detect an infestation in a hotel room.
This is a really misleadingly titled post....at a minimum it should have a question mark. It's a way false rumors get spread.
I truly hope this foirum isn't deluged with 300 some odd posts asking about each hotel in New York. It's just a waste of time. Please folks.... read through the Trip Advisor reviews. That is the best resource about specific hotels. There are some other threads on this board that have information as well.