I encourage you to try the tire liners just because you began this as a learning experience, and I think experimenting with tire liners would add value to your results. (At least for me, because I want to experiment with them.).

Be aware that you'll need to do some research about their use. I've read stories of people complaining they caused flats. Other people said it's because the sharp corners need to be rounded off. Or, that the edges need to overlap (if you try to butt them together they can pinch when the tire compresses and the liner ends move together).

You can replace those nutted axles with hollow ones, and use skewers. Google for Wheels Manufacturing and browse their axles. Choose a hollow axle that is the same diameter and thread pitch as your existing axle. And one with a length that will allow for 4-5 mm to extend into the dropout on each side, being flush (or a little less) than the outer surface of the dropout. That's like 8-10mm longer than the space between the dropouts.

In the rear, maybe not due to the claw derailer? Either way, you have to be aware of the risks. For example, the forward facing dropout is a risk. If you snug a QR lightly, a risk exists that the wheel will pull forward when pedaling hard. That wouldn't be pleasant. (For the front, less risk. But, without the "lawyer lips" to retain a loose QR, you want to be sure the front QR is always super tight.).

There was a guy on the bicycletutor.com forum saying he patented a sold axle with quick release levers. He started selling 2-3 months ago. If you can't find it, I could look. That might be designed to work better with a claw derailer.

EDIT: I found the solid quick-release axle replacements I mentioned: Axlerelease.com. It's a thread-on QR to replace the nut. It doesn't replace the axle.