Ken, copper sheet will cut the same as steel or stainless, or aluminum with Plazma. Thickness capacity on copper will be somewhat less than steel, because copper dissipates heat better than steel.
When the dealer demoed my Thermaldynamics, he sliced thru a sandwich of steel, stainless, copper and aluminum, and the cut didn't vary between materials that I could tell.
I'm presuming you are talking about thin copper sheet, .050 thickness for the sake of argument, and on that, the cut would probably be around .030 wide with my machine.
Large capacity machines do have a wider cut.
You can also gouge with Plazma, but that might not be applicable on thin sheet copper.
As far as piercing, my machine would pierce .050 sheet with a hole diameter of around .030. There might be some spalling on the back side when piercing copper, can't really say without trying it.
My advice would be to take a piece of your sheet goods to a legitimate welding supplier, and have them demonstrate. That way you'll also get a feel for how the gun fits your hand, and the work you will be doing with the machine.