Do you clip the corals off the plugs or do you leave plugs on?
Depends on the type of plug. If it's a smooth surface with no voids and the glue is nice an smooth usually I leave it together. If there seems to be any spots for smooth criminals to hide away I snap the frag off and leave a little tissue behind.
What do you do with dead areas? Maybe a dead tip or a bit of skeleton showing near the base from gluing?
I don't take corals with dead areas. If one we shipped to me and it shows up with patches I'd snap off dead tips and if it were in the middle as a patch I'd cut it off all the way past the patch, leaving only the base.
What do you use for a dip? How long do you dip for?
Bayer, similar to Mindy with a 3-stage dip to make sure no Bayer makes it back into the main tank. Flatworm Exit and Coral dip separate from the Bayer for 5 min.
Do you wear personal protective equipment? Do you touch the dip water?
Bayer I don't touch the dip and I wear goggles and gloves. FWE and CD I put my bare hand right in there.
What do you think your choice of dip is protecting you against? (<-- this is fun!)
Pretty much everything aside from egg stages of most pests. I usually am pretty harsh on my scrubbing routine on newly acquired frags. I use a toothbrush and a nylon scrub pad to work the plug and even onto the first couple mm of coral tissue.
Do you dip all corals or only some corals?
I dip it all. Cheap or $$$
How do you decide if you dip or not?
See above 
Do you examine the dip water afterwards?
Yup, just with the naked eye and a 10x hand loupe if I'm concerned.
What do you do with the corals if you see something on the coral or in the dip water?
Give them an extra eye over and decide if I am going to snap the plug off anyways. Better safe than sorry.
Do you tell the seller if you find something? (<-- they never believe you anyway haha)
Yes.
Do you place the corals in a quarantine tank after dipping or straight into your display?
Just a frag system which is inline with my main system so it's not a quarantine.
Any other details???
I like taking long walks with my wife and dog. The fish and corals don't like it when I bring them along 