I'm searching for a broad word that is summarized by sexual transgression. Is there a single word to describe a type of behavior of those that pursue rape, pedophilia, adultery, incest, or molestation?
There is no single word to cover your broad field of Sexual Predatory Behavior or even your narrowed field of Sexual Transgression.
We do call those who have transgressed so sex offenders. However, it is a very broad term which includes even some types of texting.
There is a distinction made between a sex offender and a sexual predator, because of the predatory nature of some sex crimes.
Deviant (departing from usual or accepted standards, esp. in social or sexual behavior) can be used if you are already discussing the topic, as can offender or predator, but it needs a context to be most effective.
Paraphilia is too broad a term that has been debated by psychiatrists because it includes relatively harmless behaviors (such as mild fetishes) and fantasies which need not be acted upon.
Purity carries a connotation of sexual sin, but is much too broad.
To settle the debate a bit, let's agree that your question and the list of activities in it are only culturally connected. Once we've done that, all we need to do is find a word that, in your culture, encapsulates all those things. I suggest the following: Pervert, deviant, aberrant. I'm pretty sure 'pervert' is the most likely to appear in common usage.