Replacing the battery would be a minor issue. Tires on these trailers don't last long. I just bought two new ones a few weeks ago - based on my experience, you should allot about $115/tire to replace.
I too, would be most leery of the rotten wood. If it's bad in one spot, it might also be wet in other walls as well. Just getting at the bad spots, making the repairs, then trying to make the repaired wall look decent, might be challenging to say the least.
Those 2 or 3 way refigerators are small, but not inexpensive. Being 20 years old, it's my guess that you'd wind up replacing it .
Just my opinion, but this sounds like the kind of project that would seem like fun at first, but it could "spiral downward" pretty quickly and you'd be sick of seeing the thing in your driveway.
Nick