He is at the Old Mormon Fort, which is in Freeside, on the outskirts of New Vegas. It is not a marked location on the map, just part of the Freeside area.
Arcade Gannon's quest is called For Auld Lang Syne.
The only enclave present in New Vegas is the remnants for arcade gannons special quest which can be activated near the end of for the republic part 2 or the house always wins. If siding with yes man then halfway instead of near the end.
To begin Arcade Gannon's side quest all you have to do is hire him as a companion. He will then talk to you and he will start the quest. Arcade is located in Old Mormon Fort.
You have to hire "Arcade Gannon" and a secret little quest will come along. But you have to talk him into it.
Look up the Quest "For Auld Laye Syne". You have to do a few things with Arcade Gannon in your party and he will give you the key.
if you mean rex you get it when you recruit him
Follow the main quest line or head NorthEast of Goodsprings.
Each companion in Fallout New Vegas has a quest (or two).-Arcade Gannon: For Auld Lang Syne-Cass: Hearache By The Number-Veronica: I Could Make You Care (and unmarked quest You Make Me Feel Like A Woman-Raul: Crazy, Crazy, Crazy (and unmarked quest Old School Ghoul)-Boone: One For My Baby and I Forgot To Remember To Forget-Lily: Guess Who I Saw Today (and unmarked quest Lily and Leo)-ED-E: ED-E My Love-Rex: Nothin'But A Hound Dog
If you feel like killing Arcade Gannon. He isn't able to be your companion after the quest so killing him wont effect anything or make anyone hostile.
If you mean after you have finished the main quest line then no you have to load a previous save.
I dont think you can. As you can have many quests on the go, just choose which quest you'd like to do and leave the ones which you dont want to do.
All of them except for the "Dead Money" DLC.
You have to talk to the Elder, but first you have to complete all the quest's he assigns you. Then you can ask him, and he will grant you it. :)