Can you glue vinyl cove base directly to drywall

I think you're going to have a lot of problems trying to prevent water/moisture in this garage.

As you say, improving the gutters is a good first step. It's likely that there is no drainage system around the foundation of this garage, so that is going to cause some trouble. You may be able to mitigate some of the ground water by putting a french drain around the outside, and draining it off to the side somewhere away from the garage. It doesn't need to be too deep (6" is probably enough), but there should be a gravel bed wide enough that any of the water that would drain around the walls is diverted (remember, water will follow the path of least resistance).

Ultimately, despite drainage, you're going to have problems with condensation. As the warm humid air meets the cool foundation, it's going to form condensation on the walls. This is unavoidable, since it is a garage and has a giant opening to the outside. If you build the frame around the inside in such a way that there is still air flow for drying, you may be able to reduce the effect though and at least somewhat prevent the smell. Definitely use pressure treated wood for everything here.

Is the 2" diameter pipe (connected to the downspout) the only means for water to flow off the roof?

If yes, does this current system allow the roof to drain correctly? ie Does water flow off the roof in a reasonable manner or does it build up (puddling) or does it overflow else where off the roof?

I'll then fasten the overhang to the vertical edge using a termination bar (basically a strip of aluminum with screw holes every 6" or so) along with some EPDM caulk.

This use of termination bars seems like it should be really reliable, but possibly overkill -- they're designed to help seal the EPDM on its way up a wall, and this is down.

Also the above has me a little confused, is the EPDM being terminated via being dressed vertically up a wall or is it being bent down a wall (a photo of the existing would greatly help)?