I've got an ingrown toenail and a couple weeks of go I cut off the piece of toenail out, but still have pain. Are they gonna get rid of the extra skin on the sides?

Please see your MD. Lot of things could be done 'by do it yourself' but not on human body, . Especially your own. You may be in-traducing potentially dangerous infections into you even lethal tetanus infection, please go and see you doctor or dpm.
Yes... possible. After having a chronic ingrown nail some may develop inflammation at the nail border and possible overgrowth of the skin which may need to be removed.
Injury. The sharp end of nail probably injures the skin of toe, resuled in infection or inflammation. Using cool compresses to calm it down. If no significant sign of infection, the inflammation/pain would subsided in time.