Provided it's a corded (as opposed to cordless) model, it would; the phone company generates its own power to operate the circuitry.

perhaps...

That depends what kind of phone you have. If you have a cordless phone with a charging base station and built in answering machine like most people now a days then your phone will NOT work if the power goes out. If you have one of the older push button or (OMG) a rotary dialing phone then yes, it should work if the power goes out. We hung on to our old phone for years for that very reason. I still think my mom (as anti tech as she is) was crazy for throwing it out.

If the phone has an electric plug it will not work. If it only has a cord going to a phone jack with no electric cord it will still work.

Yes, so you can call the power company.

As long as it is a land line and the phone handset is connected to the phone with a cord.