This sounds like a gross oversimplification. For one thing - even if 90 minutes is the average length of a sleep cycle, this surely varies between humans. My sleep cycle might easily be 80 minutes or 100 minutes long, rather than 90. If I were to use sleepyti.me, it might recommend that I go to bed at 11:00pm to get up at 6:30am. But if my sleep cycle is only 80 minutes long, it will be 40 minutes out in its recommendation, which means that I would wake up in the middle of a sleep cycle, rather than at end of one - exactly the opposite of what it's supposed to achieve!
In addition, although the first sleep cycle averages 90 minutes, the second to fourth cycles are generally 100-120 minutes [1] which again throws off your calculations.
In short, I'm extremely sceptical that such a simple rule could ever be effective for most people.
That doesn't mean anything. You need a real study (placebo works great you know). crntaylor is right. The sleep stages differs a lot between individuals. You need to analyze brain waves in order to be more precise (you could also try to detect movements like the "Sleep as android" app on android. But that's far from perfect.)
No, in this case it's actually not. Mathematically. See your parent for proof. And please do not bring the "a billion flies can't be wrong" argument into this.
In addition, although the first sleep cycle averages 90 minutes, the second to fourth cycles are generally 100-120 minutes [1] which again throws off your calculations.
In short, I'm extremely sceptical that such a simple rule could ever be effective for most people.
[1] http://en.wikipedia.org/wiki/Sleep#Sleep_stages