The windows will have to come out to be properly flashed after the sheathing repairs are made.

In theory, yes. I would just be worried about the manner and selection process of any contractor that put that room together. Based on his/her lack of attention to ANYTHING that is considered proper, why would I trust he/she either handled the windows properly...or picked a non trash barrel product.

Yes you are correct. The person who did the prior work was NOT a contractor at all. The prior own was a weekend DIYer who thought he could do everything. I struggle with the point that this is just a “camp”. It was rather rustic but I’ve made a lot of improvements in the 5 years I’ve owned it. I cannot do any of this work and have to rely on others. The choices of suitable service people are very slim. Strangely there was no damage or water to the inside. It all stayed in the exterior

Do you think these windows can or should be reused?

I think you could re-use the windows. The entire wall needs new sheathing and house wrap. Correctly installed flashing will be the key to prevent this from happening again. I would double check the sill plate to make sure there wasn’t any interior damage to framing.