Can a vocal cord removal successfully treat larynx cancer?

Sometimes. The vocal cord is part of the larynx, and if the disease is very limited to a single vocal cord (of two), a smaller surgery can sometimes be performed with greater preservation of surrounding tissue.
Depends. If the cancer is isolated to the vocal cord, it can be treated with surgery alone. Often times, this is done by laser surgery through the mouth (rather than making an incision into the neck). Radiation is also another treatment option.