You really do need to cover your run. No matter what birds you get, you will probably have the occasional explorer, who will not be confined unless you do have a cover over the run. You also might at any point draw the attention of a hawk.
I suggest you firstly to cover your run, it'll help you save your birds, and keep the dogs from getting free KFC.
Breeds, that I personally like and that'll be good layers,
I have to say that my hatchery Brahmas are excellent layers if you're talking really heavy! My Barred Rocks are awesome, but I have two Brahmas, a Buff and a Lt., both going on 4 years old, who lay most every day still. They lay through winter and just rarely stop except in the worst part of their molts. My Rocks are the same way.