I personally hate the term EE as it doesn't really define anything... As far as most hatcheries are concerned it's simply became the 'trendy' way to sell barnyard mixes at inflated prices...
Anyway with that said, all the characteristics of a EE are dependent on the stock they came from and what is mixed into them, and that mix is a loaded gun full of alphabet soup... If yours is only laying 3-4 months a year, it's likely just due to the mix it came from... My barnyard mixes lay quite well all summer and all winter, and almost all of my mature layers are putting out large to jumbo eggs... They might not be 'production' birds but they are hardly slacking in the egg laying department...
That is very true. One hopes their EEs generally look like Ameraucana other than the specific colors but since they are not a breed and there are no standards one can only hope that the pictures on the hatchery sites are relatively accurate to what you will receive. I don't agree on the "inflated price" comment though. An EE pullet chick from Meyer costs about the same as a Black Australorp or RIR. Maybe there are some hatcheries that inflate the price. If so, they are likely the ones selling EEs under the name Ameraucana and claiming they are rare.