Will a ct urogram show recent kidney infections or stones that passed?

No: No, stones that passed out of the body are gone. It will show any remaining stones. There may temporarily be some residual signs of the obstruction caused by the stone, however. Also, if an infection has resolved, the CT might temporarily show some residual signs of the infection, but will otherwise be normal unless the infection resulted in scarring of the kidney.

Unlikely: Kidney CT without contrast will idendify stones certainly can not if no stones are present. Might demonstrate some hydronephrosis (Swelling) if a stone has passed after causing temporary obstruction. IV contrast, in conjunction with CT can demonstarte areas of reduced uptake corresponding to ares of infection (pyelonephritis) if present. US is cheaper, non invasive & good screening study