Okay--finally got the serial number! (Husband had put the bike in the shed). It was on the left rear drop out, exactly where GTs58 said. FE34502.
Found a lookup, and it's an "original Chicago Schwinn" (a Schwinn Twinn) stamped in June of 1969! I was a toddler. Eldest sister would have been 10. June is both of our birth month.
You're gonna need a new rear rim for sure. It must have sat in dampness and got rusty as those were very durable. It is likely a 48 spoke rim and might be difficult to find a new one. but a lot of Schwinn Twinn's parted out so a good used one may turn up. Either have a good bike craftsman spoke your original hub to a new rim or try to find an exact used replacement so the rear brake and gear set is still functional.
I'm going to be late for work but I just saw this. OBG, I will measure as soon as I get home around noon! I've had my girl here fresh off Okinawa (headed for Pendleton) and she managed to convince all her brothers to come home as well, so it's been a busy 10 days. She's also regretful of her haste in dismantling the bike and super happy to hear I might fix it up. I'm in waaaay over my head with this and I'm more of a multipotentialist than someone with focused (or focusable) passion, but if I can keep my direction long enough we might get her in rideable shape! And maybe once another personal goal is met (funds-wise), I'll even be able to look into a more serious restoration (if I can find a passionate bike/Schwinn person to help me--gonna join the Boise Bicycle Project to that end, and they'd be able to help with the spoke attachment and tightening as well).
Technically, the 5 speed version was called 'Deluxe Twinn". The regular 1 or 2 speeds versions were just Twinn.
And yeah, a replacing just the rim, reusing the hub and spokes would be the best approach.
Just a reminder to remove the freewheel before removing the spokes. Some newbs will cut the spokes out first. Doh. It's very hard to get a freewheel off just a hub, well beyond most newb abilities.
Reusing the spokes should be okay. Start by putting some drops of penetrating oil on all the nipples so they can be un-threaded easier. Give that some time to soak. Some might break off when trying to unthread the nipples. Don't force it, work them back and forth a bit to get them moving. Using a correct size of spoke wrench will keep the nipples from getting rounded off. Seeing that the rim has rusted clear through, I would expect some of the brass nipples to be very stubborn. Trying to find thicker spokes in those sizes may be a challenge. DT ( a Swiss brand) 2.0mm stainless spokes may be strong enough as a replacement but the hub flanges are also drilled out for the larger spoke size and a normal size spoke head won't seat properly.
Twinns had heavy duty spokes with heavy duty spoke nipples that won't fit in a standard rim. Schwinn had heavy duty versions of the rims that had larger spoke holes. Off hand i don't remember if the Twinns had 12g or 10g spokes. You will need to measure to be sure. Measure the spoke hole in the rim too. Some decent calipers will be handy for measuring the current spokes. Drilling out the holes in a chrome steel rim will ruin the chrome finish and they will rust very quickly.
For S-7 rims, there is the standard rim for .080 spokes (~14g or ~2.0mm) that was used on all the single bikes like the Typhoon, Panther, Hollywood, Jaguar, etc.
And the .105" and .120" versions used in the Schwinn Heavy Duti and the Tandems. That measurement is the spoke thickness in decimal inches.
Spoke diameters are measured using 3 different systems, Metric, English decimal and SWG (standard wire Guage) depending on the era, market and manufacturer. Spoke lengths are in fractional inches (schwinn) or millimeters (the rest of the bike industry). Spokes are plain guage (same thickness for the entire length or butted (thicker at one end) or double butted (thicker at both ends). Mostly made of steel, either plain steel with a zinc plating or made of stainless steel. Black spokes are mystery metals, who knows what's under the coating and are usually quite prone to breaking. Some slight fudging of the conversion numbers is common.
Finding a heavy duty version of an S-7 rim will be a tough search. There were other brands of tandems that used heavy duty spokes in 36's that could work. Huffy, Murray, RBC, JC Higgins, etc. Those would all have 26 x 1.75 rims which are a bit smaller, 559 vs 572mm bsd or about 1/2" smaller in diameter. And the spoke lengths would be different too. Other sources would be older Schwinn Heavy Duti bike or a wheel off another Twinn or Deluxe Twinn. The Schwinn Heavy Duty bike was pretty much a Schwinn Typhoon with Twinn wheels. Buy one to get rims and put on regular wheels and flip the bike.
Another option may be Worksman Cycles. Makers of industrial trikes. they have hd rims but in for what spoke size? You need to measure the spokes on your bike to match that up. Again, they would be 26 x 1.75, not 26 x 1 3/4". Some people think Schwinns are heavy bikes, hah, some Worksman's are 2x a Schwinn on the scales.
Lastly S-7 rims take 1 & 3/4 tires. Any other 26" size won't fit. Those are not the same as 1.75. New tire selection is very limited as it is an obsolete size.
Wheels can be a pricey issue. Many different wheels will work. Your best bet might be to find a cruiser with heavy duty wheels and swap them onto your bike. 26x 2.125 is a super tight fit but fit if they are straight. Either way, build it the way you want.
Swapping wheels is an option, but you have a 5 speed drum brake setup. You would need to have an alternate braking system and the correct bracket on the frame to install it on.
If you want to do this, PM me a photo of the rear of the frame from the seat post clamp down. I can see if you could install a Caliper type brake. If so, you would probably still want a drum brake for the front.
I had a tandem with caliper brakes and they are not known for their stopping ability.
You could go with a single speed, 2 speed kickback or 3 speed coaster wheel as well. Those would be easier to find heavy duty spokes for. I will PM you in a little bit.