Does drinking coffee increase the severity of a gout attack?

No. Coffee should not have an effect on the gout attack. There has been some evidence that coffee can decrease your risk of future attacks.
Not common. The usual triggers to a gout attack are alcohol and red meats whose breakdown products raise seum uric acid, but individual patients have been known to have their own unique triggers. And coffee is not known to raise uric acid levels which are the root cause of a gout attack/flare.