Actually the mole was removed by the same plastic surgeon who fixed Michael's face after he returned from Sicily. The Corleones had done a favor for that doctor and he owed them big time.

I don't think Vito ever had cancer. After surviving all those shots, he lived for several years, and his only vice was sipping more wine than he used to. He didnt smoke, and he ate a lot of tomatoes and other healthy food prepared by mama Corleone so that reduced his risk.