Does the orthopedic doctor typically let one choose between a cast for a broken foot and a walking boot?

Not exactly: It depends on the nature, location, and age of your break(s). If you should not be bearing any weight, you will get a regular cast. If partial weight bearing will not cause bone movement and interfere with healing, then a walking cast will be provided. The velcro type splints usually come after a few weeks' immobilization.

Sometimes: You may be given a choice under some circumstances, but you are not always going to be given the option to chose which you would prefer.