The simple answer to one of your questions is "YES", it very well could be kidney stones. Early last year I suffered with three kidney stones for about 3 months, off and on, with blood in my urine, both heavy to the point where my urine looked completely brown or red, depending on the day, or very light and almost couldn't be seen.
After having contrast you might have, as you had, the severe urge to urinate, depending on your body. I've personally had contrast and had no severe urge and then reacted the way you did another time.
With your urine being slightly brown yesterday morning, I'd say that you most likely have a stone or multiple stones. Thankfully you had your recent Cat Scan and that will definitely tell you what the situation is FOR SURE regarding any stone or stones. From that point, you and your Urologist can form a game plan to relieve your situation. Good Luck and please keep us posted.