After a heavy rainfall I noticed that an area of the drywall was discolored and cold/soft and managed to push it in (woops). This area is usually behind a curtain so we never noticed it before. I checked outside and noticed the caulking on the side (see image 2) has some cracks so I will remove/recaulk that, but I have no clue what to do to repair the pushed in drywall/discoloration. The discolored area is maybe the size of my fist and the pushed in area is maybe just over 1cm (size of my fingernail), but it feels slightly soft/cold around it. Do I just patch this? Or do I need to replace the drywall around the windowsill?
Also, I noticed some cracks between the window and the windowsill/wall on the interior (images 4&5) and want to patch those. Do I just use caulk?