It's been a while since I looked at Mil std 810, but from what I remember,:-
If you are going to ESS each production unit before shipment then it either needs to be done before you do you tests or you need to prepare a case for not doing it for your customer to approve. It will always be easier just to do the ESS.
Storage and operating temperature, and thermal shock if applicable should be in the equipment specification.
Other things I've learnt from doing these over the years.
Always have at least one spare equipment. Take a lot of spare test cables, plugs sockets, wire, a soldering iron or two and schematics of everything you're testing. If you're taking some custom test equipment then take two sets if possible.
Take a good book, or preferably two, especially for the temperature testing. There's usually a lot of waiting time.