Doing a criminal background check on an ancestor (US, 20th century)

Factual Questions
1

According to family lore, an ancestor of mine had several criminal misdemeanor convictions (i.e. more serious than an ordinary traffic ticket but less serious than grand racketeering with a deadly weapon with intent to creep third or subsequent offense. The record most likely would have been in New York and/or New Jersey from the 1950’s to the 1970’s, with a smaller chance that convictions might have occurred in nearby states, such as PA or CT. This ancestor is now dead.

Can I do a NJ and NY (or nationwide) criminal background check easily on him? What documentation do I need (e.g. is showing that he is dead enough, or do I need to prove my relationship, or is even that not good enough to release the info?)?

2

Well, according to MY fambly lore, my grandfather (or was it great-grandfather) had two sisters (among other siblings) who went to South America (Brazil maybe?) and went into the brothel business as madams. I wonder how I would verify THAT? This was a few generations ago, remember.

The is also lore in another branch of the fambly that claims to trace some family relocations all the way back to the Spanish Inquisition, at which time the family (allegedly a respectable clan in Spain) got kicked out and went to Eastern Europe somewhere, there taking on a new name and completely burying the former family name. No idea how one would even imaging tracing that.