Aye the trophy, aka the mark of the Beast, a tattoo on the right palm grants the immunity, and general authority to move through the prince's domain. Of course it doesn't mean said Prince won't think they can get away with staking you and leaving you out in the Sun if you tick him/her off. There are also a number of perks that might be granted to you, these should be made aware to you
IC at some point, depends how recently he killed the Kindred on the list. You should have been granted everything the former Kindred owned for starters.
Alastors effectively work for the Inner Circle, there are rumours they do other duties, but in general your character should have a general idea of what he needs to do, be it track someone on the list, or some other duty for the Circle... they are in effect the Camarilla's Secret Police.. so depending on your
ST's plans, there might be reasons they have not contacted you with instructions yet.. or they may allow you the freedom to make your own hunt.. although I'll be honest.. An Alastor hanging around a city with no reason or plan, would be kinda odd. Might want to have a quiet chat with the
ST about that.
They have a tattoo on their palm, so secrecy is hard to maintain, although the general idea is they aren't walking around announcing who they are and tend to use a glove to hide the mark.. not a great idea to allow a member of the Red list to escape because they get wind of a Alastor being in the area. Their existence in generally accepted mind, and most Princes and high rankling members of a city would probably be aware at some point, especially if you are having an extended visit.
Can't see why they couldn't hold a Court office, the reason few do would be the constant duty to hunt down the members of the Red list.. well with increased resources, its all Camarilla Kindred's duty to hunt any one on the Red List.. just as a Alastor you should just know exactly who's on it, and should be getting bank rolled to hunt them down.