^^^ is correct. I failed to account for differences in moisture transfer due to reduced temperatures (and the fact that most of the world doesn't "enjoy" 90% humidity +/- most of the time). Shelf eggs do lose moisture faster for most people - but you should not wash the bloom off those.
I...
I would wash to remove the bloom then move to the fridge - not only for safety, but because the fridge will help to draw moisture out, decreasing the inner volume of the egg, and (in theory) resulting in easier peeling.
On the counter, the bloom helps protect against bacterial infestation - but...
Good question. I don't know. I can't get my hard boiled eggs to peel either - not sure if that's high Threonine levels or low. Yes, Threo makes great membranes, but the membrane adhering to the shell is the part I can't answer.
(I actually don't like eggs, so I rarely hard boil them - and...