Will I feel anything while I am anesthetized? Will I wake up during the surgery?

No, you will not feel pain while under general anesthesia. The use of general anesthetics hampers the portions of the central nervous system that controls awareness and pain.

No, arousal from general anesthesia will not occur in the vast majority of cases during the surgery. Other types of anesthesia (i.e. regional anesthesia) may just involve sedation and therefore, arousal can occur.