Is it possible to have a synchronized orbit between the Moon and the Earth, using the gravity of both bodies to hold its position (with some assistance, since the Moon's orbit is not circular). This could lead to so much: tethered lift transport, communications, deep space launches, etc.
Is it possible to have a synchronise orbit between the moon and the Earth,using gravity of both bodies to hold its position?
Yes and no. The "Yes" part: This is the Earth-Moon L1 point. The "No" part: The amount of fuel needed to stay at EML1 is huge. It's much better to be in a pseudo-orbit about those unstable Lagrange points. But now the vehicle is not staying at a single point. Many scientific probes have already been placed at the Earth-Sun L1 and L2 points. The Earth-Moon Lagrange points have not yet been utilized extensively.
This could lead to so much; tethered lift transport, communications, deep space launches ...
Not for the tethered lift transport. That doesn't make any sense. A space elevator would need to have it's center of gravity at geosynchronous altitude, making the entire mechanism orbit the Earth once per sidereal day. On the other hand, deep space launches are certainly a possibility with an EML1 or EML2 space station. An EML1 or EML2 space station / fuel depot has been seriously investigated as a waypoint to interplanetary space.
As mentioned, the L1 point is between Earth and the Moon. An elevator straight from Earth to the Moon isn't feasible, as they have two different rotation speeds, however, the L1 point is interesting for space elevators. It is one of two key point (The other being the L2 point) in making a lunar space elevator possible. From that point, a payload could be lifted from Earth to orbit using a Space Elevator, then move to the Moon, and finally go from the Moon to it's surface using such a space elevator.