Can mild diffuse fatty liver infiltration cause elevated bilirubin? Gallbladder was removed with no stones.

Maybe. Multiple causes of elevated bilirubin. Liver inflammation, hepatitis, acquired gilbert's syndrome, postoperative status, etc. Should not remove a gallbladder without stones only for a blood test finding, which hopefully was not your situation.
Yes, I suppose. But it depends on level of bilirubin elevation and rest of your liver function tests. If the bilirubin level is more than 2.0 or so or if you have other abnormal liver function tests, you should see a doctor. Difficult to tell without knowing more details. You should see a doctor anyway to control your fatty liver changes.