Do people with gout live longer?
In men, gout is associated with a higher risk of death from all causes. This would imply that their life expectancy is shorter. From a review by Kim et al. (1):


  Among men who did not have pre-existing coronary heart disease, the
  increased mortality risk is due primarily to an elevated risk of
  cardiovascular death, particularly from coronary heart disease. Also,
  an extension study of a large clinical trial among men with
  above-average risk for coronary heart disease found that a diagnosis
  of gout accompanied by an elevated uric acid level is associated with
  increased long-term (approximately 17 years) risk of all-cause
  mortality that arises largely from an increased risk of cardiovascular
  disease (CVD) mortality.


This does not rule out that some subset of people with gout (patients with less severe symptoms?) might benefit in some way from an antioxidant effect, but in general gout is obviously not good for life expectancy.