It's less than good practice to drain gutters into a French drain. The downspouts should be connected to a "tight line" system - 4" solid pcv drain pipe that is diverted to as far away from the house as possible. Maybe a retention pit or an infiltration trench or a stormwater pond. Water from your roof should be allowed to percolate back into the local watertable. It's just that you want to keep it out of your basement.

You could try and find where your roof drains ultimately terminate... if there is even a tight line system, so yes start where the downspouts go into the ground. To find the outflow look to the low point of your property. Look for a pipe sticking out of the ground in low areas of your property, if you're near a road side ditch the drain lines could drain there although many jurisdictions prohibit this practice.

If you're concerned that there is no drain system beyond the downspouts dumping water right against your foundation, turn on YouTube and find a how to video.

The ground can be saturated without being squishy, it depends on your soil, sandy, loam, clay etc. they all have different saturation and retention capacities.

If you dig a hole, like a post hole, cover it and it fills up every time it rains then you have a good gauge of your water table. Don't rule it out. I'm sure the mines near you are filled with water.

Good luck with this. Be patient, be observant and you will solve this problem.

Moisture problems can have more than one cause. The window well isn't necessarily a cause, it's at least a symptom or indicator of a condition to correct. If you are willing to do the work to fix the issues you can save a lot of money vs hiring pros. I would start with figuring out if the gutters are drained properly away from the foundation. Direct the water away from the foundation. Next, consider whether there are surface drainage issues and consider a French drain and damp proofing your foundation walls from the exterior. Next consider hydraulic pressure of water in the ground. If the water table is very high you may need to remove the flooring and cut into the slab, install a drain line around the interior perimeter of the foundation and drain it to a sump. Be ready to sweat. Don't be scared to rent the right equipment to do the job. Like a small excavator or a concrete demo saw.

If I was in your shoes I wouldn't do any finish work until I was sure the water issues were corrected.

This sounds like it should have been disclosed on the sellers report (seek attorney help). With no other information, I would speculate that this is a groundwater issue and my first guess at solving this first would be to install a sump and pump. You'll have to remove flooring and dig a hole. Hopefully, this is a relatively simple fix and once you start opening things up you'll get more of an idea what is really going on.