((Note: Read up on how a space elevator works. And orbital mechanics. Physics maybe.))
Mars has a ridously low air density. Mars weather migth seem strong, but it's barely there.
Also, have you any idea how delta-v works. You can't just knock something from an equatorial orbit in a North south orbit. A space elevator is in an orbit. It's not, like you seem to think, a really large sky scraper. It's a thing in an orbit. It can;t just fall over. Besides, with it's little weight it can't do any damage. The higher parts will disintegrate and burn up. The lower parts will land on the ground. Safely, like a feather. (Not really, but no more damage than a falling power cable. Whitout power, that is.)
We have managed to set up tethered towers 2 mile high without significant problems. Largest problem is the thing freezing. Wind is insignificant. Balloons and magnetic stabilizers keep the thing right.
The Mars series intrepretation has, like most fiction, no basis in reality. At all. ((Really, looping round the planet. What was that thing made of. Lead. And where did it go?)).
As for the Midpoint station on the moon. It'll be at lunarstationary orbit. Either Lagrange 1 or 2.
There are two lunar-synchronous points where an elevator could be placed that would be stable: the Lagrange points L1 and L2. L1 on the Earth side of the Moon is 56,000 km up from the surface, and L2 on the far side is 67,000 km up