Plumbing – How to route storm water away from the atrium

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.

If the water is coming up from below into the window wells you have a big problem. That issue means your basement is becoming a reverse swimming pool. Imagine taking a big concrete box and putting in in a lake right up to the edge of the box. Eventually many many problems will crop up that your window well issue is just the first warning of.

You are going need to take that water away from the house.

A sump pump will do it quickest and easiest, as long as the power doesn't go out during a storm (gee, somehow that seems like it might happen, plan on having it happen and take what ever measures you think necessary, like back up power for the sump pump).

Redirecting the gutter downspouts will do even better if the power does go out.

Grading the slope of the land around your house to drain away from your house in every direction will also help without power - possibly with buried plastic under the soil sloping away from the house as well as a sort of 'earth sheltered umbrella'.

Adding new better french drains around the building while performing the grading and redirecting the downspouts and adding a sump pump or two would be best of all.

But in the end, if none of the above work, I would recommend abandoning the basement entirely, filling it in with gravel and a sump pump as the water level in your soil will never allow you to have a safely dry basement.

I know one family who had to do this very thing, and was much happier knowing that their foundation wouldn't become a molding swimming pool slowly being crushed by the outside soil/water pressure. It required moving some of the HVAC and other utilities that were down there, and using the space under the floor only as a crawl space.