Solenoid would click, but starter would not engage. I hot wired it and it would start, but would not shut down with the ignition key. Once engine was killed, would not start with hot wire or ignition switch. Solenoid will not click now. I changed solenoid but still no response. How can I verify functional solenoid vs bad one and how do I check the ignition switch to ensure functionality?

With 12 volts applied across the solenoid coil, it should energize, making a connection across the large terminals. The Hot is from ignition switch and the ground may go through some safetys first(Neutral, seat switch, etc)

It shouldn't matter which end you hook into, I have see a terminal connected to 1 of them. If this coil is isolated from both large terminals, it shouldn't matter.
Have Good 12 volts hooked to 1 Large terminal, then with a good ground and 12 volts it should connect Solenoid to the Output terminal. You can jump the starter with a screwdriver(constant hot to solenoid, switched hot from ignition, again the ground can be temporarily made for the test, If that was your problem trace safety wire until found.

Thanks, I will try this the first chance I get, probably day after tomorrow. I have appointments tomorrow, and I live way out in the sticks!