After initial contact, how long should I address a professor as "Dr. Smith" or "Professor Smith"?

This is so highly dependent on particular universities -- let alone particular regions -- that it is probably best just to ask students and faculty what is the culture at their particular university.

I will give one example. At most American universities you should begin by calling all your instructors by "Dr." or "Professor", although in some cases your instructors will be neither one of these and should probably say so in response. For anyone who is either a Doctor (i.e., has a doctoral degree) or a Professor (in the United States this usually means "is on the tenure track", but already there are variations...), it seems a bit rude not to use one of these two appellations. When students call me "Mr. Clark" I assume they are just forgetting that they are not in high school anymore. But whether "Dr." or "Professor" is preferred is highly variable. At my university it seems that "Dr." is the go-to appellation: I have even heard other faculty refer to me as "Dr. Clark" when talking to students. Nevertheless I prefer being called "Professor": getting my PhD was nice. Getting a tenure-track job took place three years later, and that's when I really made it big. However at some places you call people "Professor" because you are not sure whether they have a doctorate, and for the ones who do, "Dr." is the superior honorific. Et cetera...

(I believe I learned some of this from Paul Halmos's Automathography, which I highly recommend to all academics and not just to mathematicians: to mathematicians I would require that you read it if I could! He goes on to explain more nuances than I did above.)

What you call your instructor also depends on things like their age, their gender, and honestly perhaps even their ethnic background. As a tenured Caucasian male, the desired aura of authority is already there: I don't have to do anything special to summon it. On the other hand I am still "young" -- closer in age to some of my students than some of my colleagues -- so if I met a student in a non-academic context I would not want any deferential treatment. (This is also a generational thing: telemarketers, phone company employees and so forth now call me and refer to me by my first name, and I wonder where our civilization is heading...) I am totally okay being called by my first name by any university student. Whether they are similarly okay doing so is another question, but I encourage this behavior particularly from former students and in contexts outside out of the university campus. If I were 65 years old and wearing a suit, calling me by my first name would seem less appropriate. (In fact I had to steel myself at first to call all of my colleagues by their first names, even the ones who were famous mathematicians before I was born. But that is definitely contemporary American academic culture: any of my colleagues who calls me "Dr. Clark" is signalling that they want to strangle me.)

I feel that it is especially important not to use less formal appellations for female faculty. I covered a colleague's class a few weeks ago, and one of the students asked a question, beginning with "Miss Matic said..." And my answer began "Well, first things first: it's Dr. Matic..." I then got the student to agree that that was the correct thing to say before moving on to address the question. Also I feel honorbound to stand more carefully on honorifics when addressing minorities. It is sad to me that contemporary American society has not gotten past the point where this seems necessary...but it hasn't yet.

For email, the general rule of thumb is to address the person in the same way in which they sign their emails to you. If they always sign as "Professor" or "Doctor", you should address them by the same title. If they sign informally; i.e. "Chris" or "Dr. J.", address them by that name.

It gets a little more complicated when speaking face to face, and here I tend to err on the side of formality, and use Doctor or Professor unless and until they ask me to do otherwise. As @Penguin_Knight states above, you may need to take the initiative and ask them to call you by your first name and hope they get the hint.

"Dr. [Last name]" is more common and also more versatile because it covers people with a doctoral degree but not involved with academic organization. If you do know that person does teach or you are/were in his/her class, then "Professor [Last name]" works as well.

The formality can be toned down when the other side writes something like "Please call me [First name]." Otherwise, keep using "Dr."

If you feel that it's safe to be more collegial while the other side still keeps addressing you formally as Mr. Hundt or Dr. Hundt, you can take the initiative to say "Please call me Andrew." Usually, a socially viable person (notice that social skills can be scarce among PhDs, but give it some patience and eventually this will happen) should reply and say, "Please call me [First name] as well." If not, look pass the awkwardness and keep using Dr. [Last name] until perhaps you two get to meet and be more acquainted during the meeting.