The version I have heard is "If life exists out there, why hasn't it been here yet?". That doesn't have anything to do with OUR longevity. A civilization a million years ahead of us could have colonized the earth by now. I see a few resolutions: there isn't anyone else out there (not appealing). There are civilizations out there, but it's too hard/expensive to travel between stars (also not appealing, since it means we're stuck here too). We're the first (kind of romantic, but it doesn't rule out being stuck here).
Well, it does have to do with our longevity, I think, because there wasn't human life around a million years ago (or much more recently) to know whether life arrived from elsewhere. But this kind of speculating is quite different from the kind of science being conducted now under the flag of astrobiology. It's not so much about intelligent civilizations (although they may be out there) but rather about determining whether there is microbial life on Mars or signatures of life on distant exoplanets. The logic is that one genesis of life on Earth could be unqiue; but a second genesis on Mars, too, would mean that life is most likely a commonplace in the cosmos. Same idea with biosignatures that might be detected regarding exoplanets. One solar system with life could be unqiue; more than one implies it's a commonplace.
Assuming its possible to communicate between inter-stellar planets, isn't it like that in 1000 years, the humans of earth would have technology capable to doing so to any place that had invented radio? Since 1000 years is a clock-tick of cosmological time (compared to the 4 billion years it took humans to evolve), doesn't the fact that no one has contacted us imply that there is no one technologically a little ahead of us, and hence that no place has evolved a very advanced biology?
Well, not necessarily. Back in the 1950s Enrico Fermi posed what became known as Fermi's Paradox, and it asked why there had been no contact by life beyond Earth if it was as prevalent as supposed at that time. The question, however, was based on what seems like a misunderstanding of the vastness of space and the sweep of time. We are, of course, just a small speck in a large universe, and the small number of decades when we might have been able to detect some form of alien life is nothing compared to the 4 billions years plus that Earth has been around. So that intersection of time, place and intelligence is very limited on Earth and logically would be so elsewhere.
It's kind of hard to think about this for someone who hasn't spent a lifetime trying to, but how many different possible DNA codes could there be? Like, could there be something analogous to life on a planet with a nitrogen atmosphere?
There's actually a lot of nitrogen in our atmosphere, but I know what you mean. I can answer this way: Recent NASA-sponsored research found what appeared to be the element arsenic in the DNA backbone where phosphorus normally would. The research remains unconfirmed and has been the subject of some controversy, but I find it fascinating and suggestive. If the arsenic has, indeed, replaced phosphorus in the microbes DNA, then it is a different form of life from anything found before -- since everything from microbes to elephants has DNA based with phosphorus and five other elements. Some call this a "shadow biosphere" here on Earth -- alien life, as it were, in our midst.
What did you think of (the book and/or movie) Contact?
Carl Sagan is no doubt a central figure in astrobiology, someone who taught others to think outside the box. He also did a great service by opening many minds to the vastness and wonder of the cosmos. That said, the movie Contact was certainly science fiction rather than science, and perhaps leads people to expect different results from astrobiology than it can and will deliver. I know that Jill Tarter, the woman on whom the movie was loosely based, is a scientist rather than an time-traveler, and she has worked for decades to make the SETI Institute in California into an impressive scientific endeavour. Their science is now considered sufficiently sound that both NASA and the NSF can and do entertain proposals from them -- something that was not true for more than a decade after Congress banned SETI from federal funding.