**Can a movie be good if it takes three (or two or more) viewings to understand it sufficiently? **
Yes, of course.
**Does that mean that the movie is too wonderfully dense to take in all at once or that it is so poorly constructed that it can’t be taken in all at once? **
Wonderfully dense.
However, many times the difference comes from within the viewer. I get a lot more out of Godfather 2 now that I’m 40 than I did when I was 24, and that’s largely because of maturity and experience - I’m closer in age to the main characters and some of their motivations are familiar.
**Should a movie surprise you and be about that initial reveal or should a movie be familiar and fully absorbed before one looks seriously at it? **
To be honest, they’re not mutually exclusive.
Movies are about both: the initial experience, and the depth to analyze them.
Do you learn to delve more deeply into the movie the third time through or do you just start to read things into it to justify the time spent.
Naw, I don’t think I’m reading things into a movie to justify the amount of time spent on it. I believe that some movies are good enough to justify multiple viewings and I’ll still see things on viewing 5 or 6 that I never saw earlier. Often it’s an attempt to recapture the original experience of seeing the film - this is usually the reason I would see a movie again while in its first run. For example, I saw Across the Universe twice in the theater.
Of course, many times when you see a movie again you’re watching it with somebody who hasn’t seen it. It can be enjoyable to watch their reactions to scenes - I remember a loud, girlish scream coming from a 250+ pound friend of mine when he saw The Exorcist the first time.
Does it make a difference if the three viewings are one right after the other or several years apart?
Many movies are worth seeing again 10+ years after the last time to see even if you approach the same movie the same way. My wife and I have been kind of re-watching a number of movies that we loved back in the 80’s (when we were teens) and reappraising them:
Grease: A blast to our 11 year-old eyes and ears, a horror as a parent when we realize the message of the movie is “you should turn into a slut in order to get the guy you want.” :eek: 
Ferris Bueller: Not really identifying with ol’ Ferris anymore, not really. This kid is headed into a career as a con artist or aluminum siding salesman - and he deserves it. Later events also bring about creepier interpretations of Jeffery Jones’ role in this film.
I watched Ferris a lot when it was released to video then didn’t catch it again until a year or so ago when it was on AMC or E! or something. We had a complete 180 on this movie, in some ways.
And a large number of once “good” movies that you now find are utter crap: Risky Business, First Blood (first Rambo movie), Romancing the Stone, etc. It’s good to know that the madness of thinking First Blood as a “good” movie has finally lifted. 
**How many movies do you view multiple times? Why? How many movies do you view just once? Why? **
The likelihood of a movie being re-watched is proportional to its chances of being shown on TV or my buying it on DVD. We rarely rent movies that we’ve seen already - BB and Netflix are used for new viewings, but we’ll rewatch movies if they’re on TCM, TNT, Sundance, etc.
I also have a 6 year-old, which incorporates a sense of re-watching that also bears mentioning: Her most re-viewed movies include The Incredibles, Beauty and the Beast… and, er, even less revered films such as Barbie: Fairytopia, in addition to big chunks of War of the Worlds and Jurassic Park. Last Friday we popped some popcorn, turned off all the lights in the house, and put on Twister for the first time: Sophie loved it.
What kinds of insights do you usually get after multiple viewing that you didn’t see at first?
A number of them are insights that maturity brings (Ferris Bueller, above), others are of the “I didn’t notice that” issue. Of course, the most common reason for re-watching a movie is because you enjoyed it the previous viewings so insights might not be coming.
Are you different if you watch movies over and over? Has DVDs and Netflix and cable tv changed that? Is is different for recent movies than for oft-repeated classics?
I re-watch movies most on Cable TV, with purchased DVD’s being re-watched ocassionally. I rarely rent movies that I’ve seen before.
We tend to re-watch movies that were made when we were alive: it’ll be highly doubtful that I’ll re-watch Born Yesterday (1943?) ever again but very likely that I’ll re-watch the LOTR films before I die.