How do you screw stuff into the walls?

But, be careful to get them short enough that they don't go all the way through the foam.  I don't recall how thick the wall are, but it is somewhere in the neighborhood of 1 1/4".  

We always use coarse threaded screws with a little 3M foam double sided heavy duty foam tape.  Never had a problem with it coming loose.  The screw compresses the double side tape and creates a very strong bond.  Keep in mind the vinyl clad plywood is only about 3/16" thick so you have to be careful how much weight is on each fastener.

The short ones of these should do the trick. While you can use a screwdriver to expand it, there is a tool that would make it much easier.

I did not have the tool when I used the ones I found at Home Depot. Since then, HD seems to have stopped carrying the shorter ones. Don't use the longer ones. I found out the hard way about ones over an inch long. Expanding a small one and then using a washer if needed along with the screw should hold almost any thing very securely to the wall of the RPod, even if it does not go through a frame member.

It's too bad you can't get inside the wall to put big washers on the other side of the paneling to spread the load on the thin wood.  The best you can do is to try to spread the load through other means.  We've used the coarse thread and 3M extreme/super heavy duty mounting tape that claims to hold 90 pounds per inch [https://www.amazon.com/Double-HitLights-Mounting-Waterproof-Length/dp/B07TC16SZ7] with great success.  The towel racks, paper towel holders, toilet paper holders, and so on, all are very securely fastened and show absolutely no sign of loosening after years of use.  I speculate, but will leave it to our wise engineers to opine, that the foam helps dampen some of the forces exerted on the mounted stuff and the wall, thus helping it stay in place.  But, of course, as I often am, I could be wrong.  According to Mr. Google, applying Goo Gone will dissolve the glue on the tape for easy removal, if you decide to take what you mounted off the wall.