Borderlands – Can’t create a psycho

Well it seems there is no visual indication if you only own the Season Pass:

However my BL2 did update and I quite certainly have been able to level over 50. So I would say as long as your BL2 has updated since the patch (April 2nd) and you own the Season Pass your Ultimate Vault Hunters Pack should be active.

Yes and no. Yes, that seems to cover all of the important DLC, but no, the cosmetic DLC (character skins and the like) is not included. Personally, I wouldn't be too concerned with that, as A) there's already a tonne of character customization options included in the game and major DLC; and B) it only effects how others see you (and what your hand looks like).

As for your second question, Borderlands and Borderlands 2 draws very heavily on Diablo 2 for leveling and character progression. When you beat the game the first time, you still have lots of character levels to unlock. That's playthrough 1. You can go back and play the game again with the same character on a harder difficulty. That's playthrough 2, or True Vault Hunter mode (TVHM). After beating TVHM, you can enter Ultimate Vault Hunter mode (UVHM), or playthrough 2.5, which has the same progress as your TVHM save, but ramps up the difficulty (and rewards) even higher.