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Hi, I'm Mike G and I'll be assisting you. You are correct in that a shared neutral could be a cause of appliance failure. I suggest that the microwave should be on a separate circuit direct to the panel with out a shared neutral. Most electrical circuits are on their own hot and neutr to avoid spikes and low voltage that is the reason for failure. You might consider making a circuit map of the electrical system to know exactly what is on any circuit and make troubleshooting easier. A sheet of paper oer floor, sketch the rooms, put an R for receptacles and an L for lights. With help flipping breakers mark the breaker number next to the letter. You will see it all. When something fails and outlets ahead worm, you will have isolated where to make the corrections.
That's correct. When 2 circuits share a neutral and there is a bad connection on that neutral the voltage will spike on one circuit and drop on the other in relation to the difference of the loads. On receptaclecircuits shared neutrals should not be dependant on connections to devices. They should be made up in a wirenut with a pigtail to the device. If any receptacles are wired with the push back connection, they will fail and cause your type of problem. All wires on devices need to be terminated under screws.
You're welcome. Good luck.