how to refinish and reupholster danish teak sofa  

I suggest watco teak oil. I would recommend keeping the original inner spring cushions, but it is your sofa, do whatever you want. There are so many excellent mods beyond the lousy stock configuration that really enhance the Ole Wanscher experience.

A sprung back cushion should be rigid enough to not fall through the gap in the back of the frame. That's been my experience, anyway. Have the innards of your back cushions been replaced with foam, maybe? But not the seats?

I'm all for doing whatever makes the sofa comfortable for you. You can always save the spring units if you go with foam. And even if they are lost along the way, this is a product that is still available commercially and maybe new ones are more comfortable than those made 60+ years ago--I don't know.

If you want a really comfortable sofa, consider latex foam cushions for the seat, at least. Latex is more resilient and molds to body contours much better than urethane foam. If you press your hand into urethane foam, it will compress in a dish shape radiating out from your hand at the bottom. If you do the same to latex, the foam will compress only under your hand. It also retains its resilience way longer than urethane which can start to form hollows in as little as a year.

It does cost more. But it's worth it. I finally caved and replaced my urethane foam sofa cushions with latex at a cost of $400 vs. $50-ish for urethane and am quite happy with them. You have to order latex online unless you live near one of the few places that sells it, but they'll cut it to order. Your upholsterer should ask about a trade discount; I know one place offers 10% off, or used to, anyway.

Latex foam is heavy and floppy and probably would not work as back cushions. You might be able to get awhile with a firmer urethane foam there. Some upholsterers only order big rolls of foam which I've never seen in anything but lower density/less firm. To get high density, firmer foam you have to buy it by the slab (72", 84" or 108" lengths, give or take a bit). All stuff to discuss with your upholsterer.

Oh and make sure he will not add thick polyester batting to the cushions. This is done all the time with conventional upholstery to get the look of down cushions but Danish Modern furniture should have slimmer profile cushions. Some batting is fine; it helps to keep fabric from gripping the foam and ending up creased. But only 1/2" to 3/4" at most.

Thank you both for your replies. Possibly the cushions were slipping through because they were sitting too high due to a board the previous owner had sat in the base to keep the cushions up rather than re-web. They slip behind when they are knocked from lower down like if you have your feet up for or are a small wiggly baby. Maybe the re-webbing and thinner cushions will help?

I'm not sure of your reasoning there, or maybe I'm just not envisioning what you mean by "when they are knocked from below" and how re-webbing and thinner cushions would help. Do you have photos of the sofa in its current state? Or is the photo in your first post the actual sofa?

Elastic clips are a simple solution to hold loose cushions in place, and I'm sure your upholsterer could figure that out.