Oh, no, that's not how it works. In order to get S-Rank, you must beat all 9 stages with a total time of < 7 hours. The individual stage times don't matter, but because it's usually possible to clear each stage in around the same amount of time, there are general "target" times to indicate how quickly you should be trying to beat each stage. Doing one stage faster than the target allows you to do another stage slower than the target if the difference in times matches up.
You only need to beat a stage to unlock the next one, and you can replay any stage as many times as you want. Which stages are available is determined by the player in the team with the fewest stages cleared, e.g. if three of you have every stage open and then a player with no stages joins, you'll only be able to access Stage 1 until that player leaves the team (or you beat Stage 1 to unlock Stage 2 for him).
"Fear the HUnewearl."