FMA Brotherhood and FMA 2003 are two different adaptations of the manga. Since FMA 2003 was made while the manga was still running, it eventually departs from the manga narrative and goes in its own direction for its ending. FMA Brotherhood is a more strictly faithful adaptation, made as the manga was ending. Either one can be watched independent of the other. Their respective merits are hotly contested, but just about everyone agrees that the early parts of the manga were presented better in FMA 2003 than in Brotherhood, so keep that in mind.
I'd say so. Cowboy Bebop is very good, and very approachable, so it's an ideal candidate for you to watch. Some other anime that should appeal to your tastes, which seem to be primarily in shounen action at the moment, are:
Baccano
Code:Breaker
Fate/Zero
Garo the Animation
Hunter X Hunter (at least start with the 1999 anime)
Jojo's Bizarre Adventure TV (Battle Tendency, the second part, in particular)
Kaiji
Majin Bone
Sword of the Stranger
That should be good for starters. If you ever want to branch out into other things, let us know.