How to cover gap below garage back door

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.

Since all you need to do is keep out air here (not actually insulate it), and you need it to be durable to the elements and fauna, I'd suggest filling that gap with mortar or quick-setting concrete. Once it's fully sealed up on the exterior, you can go inside and seal up any remaining gap with Great Stuff or another kind of expanding spray foam, or even caulk if the gap is small enough.

If you want to go even farther and actually insulate the sill plate and band joists from the inside (a good idea), you can do as your first picture suggests and encapsulate them with even more spray foam, or you can cut pieces of rigid foam to the right size and caulk them into place so they're airtight. The point is that if you insulate the sill plate and rim joists, you need to make them completely airtight from interior air, which is moisture-bearing and can condense on them if you insulate them without cutting them off from interior air. Once you've 100% sealed them from interior air with spray foam or caulked rigid foam, you can fill the remainder of the space with cheaper mineral wool or fiberglass batt insulation. Use only unfaced batts.

Another thing to keep in mind. You mentioned that you feel a draft when the gas fireplace is on. Gas combustion consumes air, and that air has to come from somewhere. I'm guessing that your gas fireplace doesn't have its own ducted supply of combustion air, so the air is coming from inside the house, from the biggest nearby air leak--the one you're asking about. If you seal off that air leak (a good idea), the air will be pulled from somewhere else. If you seal off enough of these air leaks (a good idea), your gas fireplace may be starved of air and cause backdrafting of any other gas combustion appliances inside the house. This is very dangerous, but probably not something to worry about right now. However, keep in mind that the tighter you make your house, the more you will need to confront this challenge/danger. The fact of the matter is that gas appliances without ducted outside combustion air supply are not a very good match for being located inside tightly sealed houses.