Correct. They only get what you see on help vindicator and help justiciar, nothing else.

They do not get the leader edge bonus (since they can't induct).

They are 'leadership' only in the fashion (as per the books) of their respective path. The book listing the "tribunal chain of command" (which doesn't specify leader) even has a note at the bottom:

"* Justiciars and Vindicators fall under the rule of the Provost in all
situations except those pertaining directly to their specialties, namely
investigations for Justiciars and the apprehending of fugitives for the
Vindicators."

And yeah, they have their own fancy rooms. Rooms I should really update sometime mainly for the list of people who have had the position... I swear, I'll do it, don't beat me!

-Ish