The rules say you can only use one Bang! per turn and one copy of each card can be out in front of you. Does this mean that I can still use two barrels or two Gatlings on my turn?
Yes, you can use multiple Gatlings or beers in a turn. These cards are not played in front of you, they are played directly into the discard pile. They are also not "BANG!" cards, so you can play more than one a turn (only the card "BANG!" is a "BANG!" card).
You cannot, however, use multiple barrels. You can only have one of each blue (or green) card in front of you at a time.
Yes, you can. Here's an extract from the rulebook:
Example. You have 2 life points left, and suffer 3 damages from a Dynamite. If you play 2 Beers you will stay alive with 1 life point left (2-3+2), while you would be eliminated playing only one Beer that would allow you to regain just 1 life point. You would still be at zero!