What exactly is Borderlands the Pre-Sequel

Per the Steam store page (emphasis mine):

Discover the story behind Borderlands 2 villain, Handsome Jack, and his rise to power. Taking place between the original Borderlands and Borderlands 2, the Pre-Sequel gives you a whole lotta new gameplay featuring the genre blending fusion of shooter and RPG mechanics that players have come to love.

Your first guess is the correct one. it is before (pre-) the sequel, and after the original Borderlands (After Claptrap's Robot Revolution expansion).

Your first guess was correct. It is (in all bar the title) Borderlands 1.5.

I can't speak for the dev's on this one, sadly because I'm not on the team, but I would say it would be for reasons to do with the storyline. What and why exactly I cannot say.

It is the third borderlands title in the series, but chronologically the events take place between borderlands 1 and 2, so it is (arguably) a prequel to borderlands 2.

I say "arguably" because while the events in the playthrough take place between 1 and 2, they are presented as being recalled during a conversation which takes place after BL2.