Can sciatica, hip pain and high uric acid in some way connected?

Yes, but not direct: High uric acid often is associated with gout, a crystal based arthritis that usually occurs in joints of the extremity, not the 'axial' skeleton. However, high uric acid can be a marker for underlying kidney disease and hypertension, which often accompany obesity that frequently aggravates low back injury and may contribute to degenerative spine disease. A pcp evaluation should be obtained.

Possibly: Gout is known to cause some discs in the spine to calcify. I would have your doctor order some X-rays/MRI to be sure, but its not too common.