SO what I need is to have the armature set to either dynamic of rigid body. But for some reason, when I set it up this way, the whole mesh and armature falls through the floor and disappear from the scene.
The common Way is to set a Cube around the Armature and parant the Armature to that Cube. The Cube works as Bounding Box, you set it to Dynamic and Invisible, while the Armatur itself plus the Mesh should either have No Collision (best) or be Ghost, as not to interfere with the Bounding Box Physics.
In the blend file that you attached, you have two boxes (I’m guessing for collision). One is Dynamic, and the other is Rigid Body (and hidden). Unhide the hidden box (Alt-H) and delete it, and use the visible box (which is what your armature is parented to) for collision. Simply deleting the hidden box will make the blend file work. You can make the remaining collision box Rigid Body if you want, just make sure that multiple objects aren’t overlapping each other when you start the game engine. Otherwise, you’ll get this ‘shooting’ effect.
Okay, select nothing. Press Alt-H. A box will appear - you will be able to see it in the scene. Then delete it. At least, there was one in the one you sent me… Okay, here’s the altered file. Couldn’t upload it to BlenderArtists for some reason. Here’s the link.