I have 2 questions about HID lamps
When I test lamps, I usually don't let them do a full warmup. This is so the lamp remains cool enough to touch so I can unscrew it without getting burnt.
This made me wonder if unplugging the lamp before it has fully warmed up will shorten its lifespan.
I'm also curious to know how long it takes a fully warmed up lamp to cool down enough so I don't burn my hands

If its just occassionally just to test or so, it won't be a problem. But doing that repeatedly will cause problems with some lamps.

Here’s what the GE lamp guide says if it’s any help 

It's always best to let lamps run for a few hours. Obviously if you are only testing a lamp from the collection, as opposed to one in service, then it's not as important, as it's unlikely to reach EOL, but not even letting a lamp achieve full warm-up is not going to do it any good.

Personally, I always let a lamp run for a minimum of 4 or 5 hours. Most low wattage lamps will cool enough to be handled after half an hour, maybe even less, but some lamps, particularly LPS, don't like being handled too soon after switch-off, as the sodium metal is still liquid and can run about inside the arc tube and possibly get behind the electrodes, which will definitely shorten life.

In short, the longer the run time, the better. But at least let the lamp warm up fully.