Does gout cause a fever or headaches?

Possibly. While not all cases of gout cause fever and headaches, it is seen in some patients. This is especially important to note as the "gout" attack may not be gout at all and possibly a infected arthritis or septic (infected) joint. These require medical attention. A headache with gout is generally not associated unless fever is involved.
It can. A gout attack can give you a fever, headache and an ill feeling all over your body.