From the context, I really think “I smiled understandingly” make more sense to the readers. But if you really insist that “I smiled understandably” is exactly what you mean, that is fine. About “the”, I asked several of my colleagues, and they all said either “a” or “the” should be used. It is funny that I asked my 10-year-old son about it this morning. He said either “Is it a dentist’s office?" Or “Is it the dentist’s office?" is correct. I asked him why “Is it dentist’s office?” is incorrect. He said: I don’t know. It just sounds weird. Haha
