If I undergo surgery, will the surgeon know if all the cancer has gone straight away?

No. The surgeon will have an impression but won't know until the path report is available.
Not directly. Following surgery, the tumor is sent to the pathologist to carefully evaluate the cancer and the surrounding tissue, with our goal to have "clear margins". While this may give us an indication of local control, most cancers have the potential to spread throughout the body; therefore, cure may require treating the whole body (systemic rx), via chemotherapy.