Is it possible for a carotid surgery to cause a stroke?

Yes. The most common side effect but should happen no more then 2% of the time. My stoke rate is <0.5%.
Yes, but rarely. Although strokes can occur from carotid surgery, the risk is very low. The surgery is recommended when the risk of stroke with leaving the blockage in place is much higher than the surgical stroke risk. Overall surgical stroke rates should be less than 6% in symptomatic patients (NASCET trial) &amp; less than 3% in asymptomatic patients (ACAS trial) to justify the operation. Stents have higher strokes.
Yes. Although strokes can occur during carotid surgery, overall they dramatically decrease stroke risk in patients with carotid disease compared to medical therapy. Overall surgical stroke rates should be less than 6% in symptomatic patients (nascet) and less than 3% in asymptomatic patients (acas) to provide an overall benefit over medical treatment.
Yes. A couple of ways. To repair the carotid the vessel is clamped off from circulation. The other side's carotid and the vertebral arteries try to pick up the slack. Some surgeons use a shunt, which detours the blood around the clamped area. However, this procedure may dislodge plaque also causing stroke. Studies have compared the techniques with no difference in outcomes.