Would a fitness professional recommend totally eliminating fat from your diet

No, you need fat in your body. The only reason people become overweight is that they eat too many calories and don't work it off. Then your body has an excess amount of calories and stores it away, causing fat build-up. The reason you eat more healthy when trying to lose weight is to provide your body with the nutrients it needs, but to also not add to the fat build-up. Exercising gets rid of the fat build-up, causing you to lose weight and fat. But no, fat should not be completely taken out of your diet, regardless if I'm a fitness professional or not.

No a fitness professional wouldn't recommend totally eliminating fat from your diet but reducing it to a small once a week or so

Cis the most source of vitamins