Welcome to the forum.
Your trailer can use any 12volt battery. How big and of what quality depends on how long you want it to last. A basic and cheap group 24 12v is the minimum. Larger group 27 and group 31s are better and will last longer between charges (although also more expensive and larger). Read this for a better understanding Forest River Forums - Downloads - 12 volt side of Life
Roof and ceiling repair:
careful about taking on a "project" like this unless you are getting a screaming deal and know what you are doing. Water is strange and the leak could be from somewhere that you don't expect and can't easily identify. You will have to find the source and repair that or the same problems will occur.
I can't help much with the materials (luan, insulation, etc) you need to do the repair as I have never done it. Others have though and hopefully they can chime in. Here's a search list from the site of other threads that might help, too. https://www.google.com/search?q=ceil...iverforums.com
There are a lot of products and methods out there. There's more information in our Library
for further reading about recommended cleaners and other items for each type of roof, etc...
Forest River Forums - Downloads - Awnings, Roofs and Exterior
Here is what I do:
I like Simple Green. First, its good to ID the type of roof that you have: Rubber roof (AKA “EPDM”) membranes are smooth and will be slippery when wet. TPO roof membranes have a texture like an orange peel.
Basically, the roof needs to be cleaned with a non-petroleum based cleaner. While you are up there, closely inspect the seals
for cracks and use Dicor LAP sealant to touch-up (leveling for horizontal surfaces and non-leveling for vertical surfaces).
I have only ever used two types on the roof of our 233S: Murphy's Oil Soap or Simple Green All-Purpose Cleaner. The Simple Green really did a good job - better than the Murphy's.
My roof is TPO so double check these can be used with EPDM if that is what you have.
I like Wash Wax ALL Cleaner from Aero Cosmetics. I will use water/soap to get the heavy road grime off where needed but the Wash Wax All does a good job for 95% of the trailer. The key is to take your time even if you need to plan on a couple of days. Spray on in a small enough area where you can immediately wipe dry. Do not let it dry on its own as it will leave streaks. A box of disposable rags is nice to use.
I like LA's Totally Awesome. The process I use is to dilute the concentrated Awesome, spray it on the underside (make sure to wear safety goggles and old clothes), climb on top of my camper and spray it on the topside of the awning, roll up and let it sit for 30 minutes. Unroll the awning and wipe off with a mop and water. Rinse top and bottom with a hose when done cleaning. Let dry.