Hi Kristine and welcome to the boards. Drywall will weaken once it get's wet and dries out. I would ask the landlord to cut out the rotten drywall and replace with new.

No,this cannot be repainted after it gets wet,this will need replacing.

It seems to me like the landlord would want to remove the drywall to dry the wall out and remove the wet insulation. Any comments on that from the experts???

Devildog

There is not only possible trouble with weakened drywall, but the more serious problem is that if the space is not dryed thoroughly, mold will grow there which will destroy his investment and cause health problems. The drywall problem will only be cosmetic