Would a ct without contrast of the abdomen and a urine culture rule out any kidney problems?

Not necessarily: Many early kidney diseases may not be picked up by the tests you described. All of your health information, lab results and imaging studies need to be evaluated in the clinical context.

See answer: Ct of the abdomen without contrast will identify abnormal calcifications and hydronephrosis but will not evaluate completely the collecting system and the functioning renal parenchyma - intravenous contrast is necessary for these purposes.