You can jumper around both the neutral safety switch and ignition switch to eliminate them as the problem. However, since this appears to be an intermittent issue, that may not resolve anything during testing. Maybe next time it is acting up you can do the jumpering and see what changes..

Yes, solenoids can go bad.

Yes, they can be replaced.

However, it might be time to just up and buy a nice, shiny all new one and stop farting about with something that is old. Unless it's some sort of all original car that is and if it is rebuilding your old solenoid might be an idea to consider. No kits that I know of to do that, however, it can be done.

Well, dang, they do make a rebuild kit for your solenoid, assuming it is the original starter. Rock Auto has them. I;m used to old flathead mopars and no one makes kits for those. You have to hand wind your own, lol.