Would a family doctor do tests to see if a young female (24) had joint damage from back cracking? Or do they wait to see damage as you age?

No evaluation needed. It may be the way you are built or that your musculature is just tight. Cracking noise or sensation can be from several origins ranging from the small joints of your back "popping" like the cracking of your knuckles to tight soft tissue structures giving the same sensation when stretched or moving over one another or another musculoskeletal structure. It is not a pathologic or bad thing.