It really depends on the hen. Most lay between 6-8 months, while others can take up to 12 months! There are many factors to consider also, etc. lighting, feed, conditions, how many togehter, just to name a few. I wouldn't worry to much as she will lay soon enough! Good luck :)
I live in upstate NY and it gets mighty cold and a lot of snow. I have a couple of coops built 10x16 with electric in one. I use only hardware cloth so nothing can get in or chew it's way through. If your building please don't use plain old chicken wire. Minks, raccoons, and all those...
Ee have the all over color like yours does, where's true Amerucunas are either, black, blue, silver, white, wheaten, blue wheaten, etc. they all have the muff that covers most of their face. EE Can have the muffs too