Sentence 2 is ok. But the patient does not "do" the test. A doctor or lab technician does. So I would prefer, "Which doctor referred you for this test?" or "Which doctor referred you to have this test done?"

I am not sure what is wrong with the ways you said it,unless you mean "The doctor is available BETWEEN 9 and 13". The doctor can see you at 9 and 13, is also possible. The second sentence is fine.No need to change it.