Would you start allopurinol for a patient with uric acid kidney stones who does not have hyperuricemia or hyperuricosuria?  

Definitely not! The main risk factor for uric acid kidney stones is neither hyperuricemia nor hyperuricosuria; it is hyperaciduria. Typically uric acid stone formers have a urine pH below 5.8. Raising urine pH into the mid 6s will not only stop new stone formation and existing stone growth; it will ...

I do not start allopurinol or other uric acid-lowering therapies in the absence of hyperuricemia. The most likely cause of the uric acid stones is aciduria, which I would treat with dietary changes, possibly with citrate or bicarbonate, as we would want the urine pH to be between 6.0 and 6.5.