You need to measure the voltage going to the coil. Too much and/or too little can cause problems and “kill” the coil. It seems you have a number of electrical issues, perhaps the ignition switch is bad. Likely the electrical issues are related so if you find the cause for one, you might cure the others too.
I’m also suspect of the new distributor. Where did it come from? Is it properly matched to the car? Could there be something that got wired improperly with the installation of the new distributor.