Re: Fixing Outdoor Receptacles
Like I said, the panel hasn't been labelled to identify those outdoor receptacles so it will be near impossible to know if I've turned off the correct circuit breaker before proceding to safely dismantle the receptacle.
To clarify some of the previous posts: So much information! By now you probably have the job all finished up!
Someone pointed out that you can turn off the main breaker to your house, which will "power off" everything in the house. That's the safest way.
You seem concerned about getting shocked. Wear thicker, dry (leather) work gloves, a layered jacket (not a windbreaker), stand on a dry piece of plywood, and tape up your screwdriver and pliers handles. Only use one hand at a time (meaning, put your other hand in your pocket). That way it would be extremely difficult to get a shock, no matter what happens with your tools. If your main breaker is off, this is unnecessary.
From your pictures, it looks as if the socket mounting screws are so badly rusted you may not be able to get them out easily. Try a vise-grip if the screw heads get stripped out. If that doesn't get the socket out of the wall, you may have to pry, cut, bend or grind the screws or mounting bracket to get the socket out of the wall. Try not to damage the box that's mounted in the wall - it could be hard to replace it.
It is not likely that both the front outside outlet and the back outside outlet are both powered by the same breaker. It's possible that the rear outlet is powered off of the bathroom GFCI, if you have one.
Don't reuse the sockets. Replace them with a better quality socket, not the cheapest. Use a weather-resistant cover over the new socket.
Wiring the socket: Black wire to the Brass screw side, white wire to the silver screw. Bare wire to the green screw.
PS - In my house my outside outlets are powered by breakers that also power the inside outlets near the adjacent outside outlet, and they were not GFCI protected - it's an older house. My living room breaker powers my front outside wall outlet, and my family room breaker powers the back porch outlet.