Star Trek did have pirates by name. Through TAS's The Pirate of Orion, which Roddenberry himself called one of their better first-draft scripts.
The TNG crew did "fight' between themselves. In the episode The Child, Worf position that pregnant Deanna should be taken to sickbay and have her son aborted wasn't popular. And the command crew had a spirited discussion about the prime directive in the episode where Data contact a child on a "primative" planet in Penpals. Doctor Pulaski didn't exactly get along with Picard.
While she was polite to him, it's pretty obvious that Janeway had no particular use for Chakotay.
In terms of soap opera, there were elements of such in the show, from time to time. LaForge's love life would be one example.