1. Only staging area, you need special effects to pull stuff off other players.
2. No, they stay where they are unless a card effects says otherwise.
This is correct and here are the sources:
1. First, each player has the option to engage one enemy in the staging area. [Core Set rules, p. 16]
2. The first player compares his threat level against the engagement cost of each of the enemy cards in the staging area. [Core Set rules, p. 16]