Should everyone get a second opinion for a diagnosis of stomach cancer?

No. Regardless of the type of cancer, it is critical that every patient that requires surgery have a crystal-clear understanding of their treatment. As a cancer surgeon, I feel that communication is as important as the operation itself (and usually takes longer). If a patient is not comfortable with what they have heard, a second opinion from an independent doctor of the same specialty can be helpful.
Not necessarily. A second opinion can be a very useful visit, especially if you are not totally comfortable with your physician team. Stomach cancer is involved enough that even if you aren't getting a second opinion, you should meet with a surgeon and an oncologist. In most situations, these physicians work together to provide you the best care possible. Other specialists might also get involved with your care.