Water problems and stud repair

So I started tearing apart this little bonus room to make it into a library and once I pulled out the carpet I found laminate and the underlay to the laminate was soaked. Unfortunately it looks like this job became a lot bigger. But my first step is to fix the leaking issue. Both leaks seem to center at the corners of the room which is the gutter downspouts are and it seems like the water from these has been just running into the room so I added these to get the water away from the house.

Should I also add a french drain and waterproof brick paint along this wall?

Also is stud repair as straight forward as the internet makes it seem? I am assuming I should replace the wall insulation in this area.

Any wet insulation and drywall should be cut away and replaced. Studs that are a bit wet can be bleached and allowed to dry with a fan; replace anything that's saturated. Just use common sense when replacing any load-bearing framework: replace small sections at a time, where possible add reinforcements before cutting away old material, etc.