only for some, you can go back to the pitt, point lookout, and I think you can go back to were you do broken steel but i haven't tried yet.
a quest
you don't.
just continue on with the main quest and you'll be able to do it.
Rivit City
get 10 demo
The quest Head of State is given by Hannibal Hamlin at the Temple of the Union.
yes
if you mean rex you get it when you recruit him
I personally recommend Fallout 3 to any other Fallout game. Fallout 3 has a more organized and in my opinion interesting storyline. And you. And you can also keep the game going after you beat the main quest on Fallout 3 if you buy the DLC "Broken Steel."
Sprocket's quest is to take a "spare part" to a Fallout near the Fallout compound in Laputta Station without going through Funkeystown. You can do it by going from Nightmare Rift to Daydream Oasis to Laputta Station.
You buy the "Operation: Anchorage" addon.
yes it is a reward from euligy Jones in an unmarked quest