Wow, Lots of information, and some of it is actually correct. Here's what you have, and what you need to replace.
You need a 26 x 1 3/4" S-7 chrome Schwinn rim. They come "stock" in 36 spoke hole, NOT 48. You want to use a .080" standard nipple size, or a .105" heavy duty spoke size. Either rim can easily be drilled for the .105 heavy duty spokes that were used on your wheel. Try to carefully disassemble the wheel, do not round off the soft brass nipples, use a correct sized spoke wrench.
The absolute best advice you have already been given is to remove the freewheel from the hub (with a proper remover) BEFORE you remove the spokes from the rim. I repeat, remove the freewheel with the proper remover "before" you remove the spokes.
You have a Union Brand rear drum brake hub, and it does look different than the more common Atom rear drum brake hubs. They will both work fine for this project, just clean and lube the moving parts. It would be a good idea to sand with crocus cloth both the brake shoes and the rust off of the inside of the drum for better braking.
I have no idea what your wheel building experience is, so I "will pass" on giving any advice. If you have a buddy with wheel building experience, maybe ask for his help. Just make sure you note the spoke cross pattern number "Before" you take the wheel apart.
Good luck with your project.
John