Originally, I was going to only install 1 layer of 5/8" drywall and have spaced all the electrical boxes 5/8" from the studs. However, now I am thinking about doubling the drywall on all walls or possibly only on the interior walls.

1) Would it be of any benefit to only double the interior walls? I believe it would help to reduce noise from the nearby laundry room.

2) Since my electrical boxes are already installed 5/8" from the studs would I simply use electrical box extenders for the added layer of drywall?

3) If I double, I probably would use both 5/8" and 1/2" drywall. Which size drywall would I apply first?

4) If I double, would I need to mud and tape the first layer?

1) The general advice is to do everything if you want the best sound isolation. It is clearly preferable. The sound from that laundry room will be in the studs, go up the ceiling, and come down. Will there be benefit for just doing the one wall? Probably. The problem is that if its not enough, its often hard to do the ceiling later. By the way, I would be more concerned about the ceiling than the other walls if you are worried about a specific sound like a laundry room.

2) I would go extenders if your drywall is already up. New boxes aren't that expensive though.

3) Gut feel, 5/8" first, but I can't think of a reason it matters.

4) Not really. You do want to seal the seams, however. You can do that with caulk instead of tape/mud. Also, if you tape/mud, you don't have to float it and make it perfect, just get a good seal.

If you are really concerned about the laundry room noise, you might go to the root of the problem and work on the source. Put your washer and dryer up on sub-dudes or other isolation devices! I'm only partially kidding. It's easier to quiet the sound at the source than it is to keep it from coming through the walls.

Originally, I was going to only install 1 layer of 5/8" drywall and have spaced all the electrical boxes 5/8" from the studs. However, now I am thinking about doubling the drywall on all walls or possibly only on the interior walls.

1) Would it be of any benefit to only double the interior walls? I believe it would help to reduce noise from the nearby laundry room.

2) Since my electrical boxes are already installed 5/8" from the studs would I simply use electrical box extenders for the added layer of drywall?

3) If I double, I probably would use both 5/8" and 1/2" drywall. Which size drywall would I apply first?

4) If I double, would I need to mud and tape the first layer?
Yes, double and put green glue between the layers. http://www.greengluecompany.com/

Some things that will help Laundry noise. It is much easier to control noise at the source rather than trying to block it after the fact. Options (can do some or all):
Replace the washer/dryer with quieter ones.
Use vibration dampening mounts (replace the feet with these).
A rubber pad under the appliances.
A new sub-floor for the laundry room (green glue and/or other techniques).
This should help on structure born mechanical noise.
The other source of noise is water running down the drain pipe. Depending on where the pipe runs (towards or away from the theater), this might be an issue.