Is a 7 degree slope for a roof too small in a rainy area

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.

It will work as well as the previous path has. Building the slope into the path means you don't have to worry so much about the base - one of the main reasons for a good crushed rock base is to aid in draining. Also, this is not a large slab area - it's just a narrow path, so going with a packed dirt/clay base is fine.

In the worst case, you'll shovel it back into a mesh to separate the pebbles from the dirt, and re-lay it out later, but I expect if you follow the design of the existing path you'll be just fine. It'll match better as well.