Correct. Some breeds are supposedly better winter layers but if you ask a dozen keepers they'll probably name a dozen different breeds.
EEs aren't really high production birds. I find them fairly reliable for my purposes, but because they're not bred for anything other than variety in egg...
Not unusual for them to stop laying completely post molt and until daylight length starts increasing. 1st of January isn't all that long after winter solstice, so doesn't surprise me that only 1 started up and then didn't continue consistently.
You'll probably start seeing eggs in Feb, maybe...
Unlikely, but right now I can't think of another reason, other than a predator stealing eggs (but 7 months straight of stealing w/o evidence is unlikely too).
My guess is they will lay in spring, at least most of them will. Very rarely you have hens that stop completely while still "young" (I...
New additions are stressful, yes, especially if the rooster is rather randy. I wouldn't anticipate a 7 month strike from that unless he was being really rough with the girls though.
As far as diseases, can't think of any communicable diseases off the top of my head that would leave birds...