A space elevator makes no sense on the moon. First of all the orbit is unstable and you would need constant station keeping.
Also, it turns out that if there is no atmosphere you don't actually need an elevator, you can just use electrically driven propulsion without an elevator. You just use a rail gun to shoot stuff into orbit.
> when they get in trouble, there is no help.
But there will be at least one professional doctors and likely a highly advanced medical facility.
> Mars settlement is only viable when we have a spaceship factory on the moon, nuclear engines, and we could send the equivant of a large marine research vessel to mars, so 10,000 tons usefull payload, every month.
You would first need to prove that building such infrastructure on the moon itself isn't far to un-practical and more expensive then going directly from earth.