Foster Brake + Through Axle
I'm sure you can find exceptions, but most or at least many trainers can be made to work with nutted axles. The answers to your questions:
1. I think you mean quick release skewer and not through axle, and not practically because it would involve transplanting in a hollow axle, and hollow axles with the thread size for coaster brake hubs don't really exist.
2. Yes you could do this, but it's not very straightforward, because it would need to be spaced right for the frame, and it would need to be a singlespeed wheel to play nice with the chain that the bike probably already has, and singlespeed QR wheels are a rare breed. So you're talking about buying a standard-axle singlespeed road wheel, freewheel, tube, rim strip, tire, quick release, hollow axle, and some hub spacers, then chopping the axle down to fit a 110mmm spaced frame and installing it in there and getting the spacing right. That all is possible but probably not the simplest path.

Here's a video that shows installation of a nutted axle on a Cycleops.