Can hyperuricemia lead to rheumatoid arthritis?

No, If you would like to have acloser look at the effects of Hashimoto's disease go to the link (Thyroid conditions - Hashimoto's disease) below.

Having hashimotos disease doesnt lead to to rheumatoid arthritis however having one autoimmune disorder puts you at risk for others. So If you have hashimotos your more likely to get RA then someone who doesnt have hashimotos. (I have both)