In general, stone analysis (which requires straining the urine) isn't done on the first stone, only on recurrent ones.
The pain happens when the stone is in the ureter, which carries the urine from the kidney to the bladder. The muscles of the ureter spasm, causing the pain. If the stone pops back up into the kidney (the central urine space, called the pelvis) the pain stops, but the stone is still there and can try to travel down the ureter again. If the stone is passed into the bladder, the pain also stops - the bladder is large enough that there's no obstruction.
A CT will show the stones, and no contrast is given (because contrast shows up white like the stones) but one CT scan is equivalent to 50 X-rays in terms of radiation, roughly.
People used to be advised to follow a low calcium diet if they had kidney stones, but the evidence now shows that advice was misguided and more harmful than helpful. The diet varies depending on the type of stone, but for the most part, staying hydrated and eating a low-salt diet are key.
Katie, Mom to two boys
Retraining my dopamine circuits thanks to David Kessler, MD.
Jonathan: Halloween 2004
Alex: A smidge past Groundhog Day 2007