Rockets carry oxidizer along with fuel, so the composition of the local atmosphere doesn't really figure. Also, your factors are way off; Martian gravity is a bit shy of 0.4g, and lunar gravity a bit proud of 0.1.
Finally, we don't need to land on the moon, or Earth, or anywhere else, to validate a method of landing on Mars, because we already know how to land on all of those things. We've done it plenty of times before, in all cases; the difference between doing it manned and doing it unmanned is purely one of mass.
We've only landed light things on mars, curiosity is the heaviest at 889kg, and the methods used don't scale easily.
Anything that can sustain humans for an extended period of time is going to weigh a lot more. EDL (Entry descent and landing) is a solvable problem, but it's not a solved one. You can find lots of recent work on it by googling that term.
Not to disagree that we can test on Earth better than the moon. Mars has a light atmosphere, but it doesn't have no atmosphere.
Finally, we don't need to land on the moon, or Earth, or anywhere else, to validate a method of landing on Mars, because we already know how to land on all of those things. We've done it plenty of times before, in all cases; the difference between doing it manned and doing it unmanned is purely one of mass.