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The bird you have is probably some sort of production type Rhode Island Red. It's hard to say, but more than likely the are great layers. Meat chickens, are really any type of chicken used for meat. However the cornish x's are used only for meat, they are processed at 6-8 weeks of age. Which is why people call them meat birds or "meaties"
Any chicken can be used for meat, so do your homework before venturing into that area. You have Dual Purpose breeds (Buckeyes, Delaware, Plymouth Rocks, ect.) and then you have the crosses ( Cornish x Rocks, Colored Broilers, or Naked Necks)
There are many different avenues you can take, so read up on them. To answer your other question, yes you can eat the layers after they are done laying. They amazing dishes, my favorite being "chicken noodle soup"