"A fit of X" means an episode, or instance of that emotion. So "a fit of professional jealousy" is a particular moment where the person was overcome by jealousy. Similarly you can say "a fit of rage", "a fit of passion" and pretty much any other strong feeling:
In a fit of spite, he took her car and drove it into the ocean.
"(A) sort of a ..." is a standard phrase to say something is like something else, or that it is similar, but not quite the same:
He was sort of a good student, in that he got good grades, but he showed little interest in what he was studying and promptly forgot everything at the end of each semester.
In many cases "a sort of" is simply a figure of speech used when the speaker wants to soften or qualify an otherwise harsh comment:
I wouldn't say the politician is stupid, but he's sort of an idiot.