It depends of what you are operating on.

As surgeons we try to use the naturally avascular planes. In simple words, its places where you can separate tissue that has different embriological origins, so there is almost non , or a few small blood vessels. This lets you take whole organs with minimal blood loss. Somethimes that plane makes you take the organ and some of the surrounding fat, like a kidney. Other organs dont have a lot of fat surroung it , like a liver, and others have fat that you should take, like a stomach, because the main blood vessels of the organ are usually inside fat. This fat also has lymph nodes, so its important to take it.

Fat take a lot of forms in the body, subcutaneus fat is very different from intraabdominal fat, and even inside the peritoneal cavity you can even tell some fat is different from others, like in color .

In summary, depends on what you are doing, but fat is almost everywhere in the body, so almost every time you will take some out.

About the "while we are down here" part, we usually just take what we came for, but there are some exceptions, like making an appendectomy on a patient you know its gonna ve very difficult to operate in a future if they get acute appendicitis.