Would a Lunar space elevator only be stable if it passes through Earth-Moon L1 or L2? I understand that space elevators are extremely hypothetical (bordering on pure fiction), but they are supposed to be theoretically possible.
My questions is about what longidtudes a lunar space elevator could be located at. According to Wikipedia:

There are two points in space where an elevator's docking port could maintain a stable, lunar-synchronous position: the Earth-Moon Lagrange points L1 and L2.

Considering that space elevators are supposed to be held in place by centrifugal forces, I don't see why points of gravitational equilibrium should be necessary. Could it be that the lunar space elevator described in Wikipedia is primarily held in place by gravity, not the Moon's rotation? If so, does that mean a primarily centrifugal space elevator at other longitudes is impossible (perhaps due to stationary orbit being outside of the Moon's Hill sphere)?
In either case, space elevators at nonequatorial latitudes are supposed to be possible. Does this mean two or more space elevators at the same longitude, and using the same Lagrange point, would intersect?
 A: For a space elevator to function as intended it needs to achieve force balance. One part needs to be counterbalancing the part that is attracted to the body it is standing on. That means that there has to be a point where locally the gravitational acceleration is equal to the centipetal acceleration. However, this does not have to be a Lagrange point - for Earth, it would just be the geosynchronous orbit. 
In the case of an equatorial lunar elevator it will pass through L1 or L2 since the lunar rotation is synchronous. But had the moon been rotating it would have just been enough to pass the selenosynchronous orbit and have enough counterweight above. 
A non-equatorial elevator will be slightly curved, since the lower part will be attracted towards the centre of mass of the body and this is a nonparallel force on the cable. I think the same will happen with a longitudinally displaced Lunar elevator that does not pass L1 or L2. By continuity there will be some point where the inward and outward forces balance each other, but it will not be a particular named point. 
