Consider checking in the Breeds section. Some good egg layers and dual purpose breeds are Plymouth Rocks, Rhode Island Reds, Golden Sex-Links, Orpingtons, and Leghorns. Meat birds are usually Cornishes or Cornish Crosses
I am no expert in chickens but she may have a calcium deficiency or mites. If you dont supply them with grit, start doing that and see if that helps. If not, use a mite treatment. If all else fails, take her to a vet.
The egg may have gotten stuck inside of her. This article may help: http://thechickenkeeper.wordpress.com/2010/05/04/egg-bound-chicken-treatment/ Otherwise, since it is her first time, she may just be taking her time.
Currently, it looks like a pullet (hen), but I am not 100% sure. I would wait until all of its feathers come in to be sure. If the vent is a slit, not shiny and sometimes, has circular bumps around it, its a hen. If it is shiny and circular, its a male.