During many intensive cancer surgeries, can cancer likely travel through veins and/arteries to different parts of the body?

Yes: Cancer is unique in that it is immortal and has the capability of metastasizing to different organs. The latter is due to production of matrix metalloproteinase allowing blood vessel invasion and breakdown of cadherin to small e-cadherin which promotes spread. What spreads to liver only grows in liver and what is in bone stays in bone based on tumor membrane specific proteins.

Yes and No: Yes cancer cells do get into the blood vessels either spontaneously or during surgical manipulations, yet they rarely survive their trip in the circulation and are eliminated by the immune system or fail to establish any colonies under normal conditions. So surgery is generally considered safe from this theoretical point of view.