Realistically, the middle of the ocean is one of the worst places to build a space elevator. This can be offset by the financial and regulatory advantages of not dealing with a country.

To make the elevator feasible, it would be tethered to the ocean floor, with a fixed platform just above the max water level to hold the elevator machinery. This platform would then be tethered to the city for commerce and personnel transfer. The upside of this is that power could be harnessed from the ocean floor via geothermal plants, allowing the platform to function independently of the city for short periods of time in case of emergencies.

The physics for a space elevator are fairly straightforward. A high tensile cable is run from the equator to a point above geostationary orbit, more than 35,000 km or so, the exact distance depending on the mass of the counterweight. A counterweight at this height is designed to offset the effects of gravity, keep the entire cable straight and under tension. The logical (and only practical) place to put a high orbit station is at the centre of mass of the cable and counterweight, ideally at 35,000 km, but this isn't always the case. This way any additional mass does not affect the system and throw off the tension of the whole. Within certain limits, practically anything can be built at this height without unduly stressing the system as a whole.

For practical purposes, a station at this height would serve primarily as a spaceport/warehouse of sorts, storing shipments for various free orbiting stations until they can be picked up (at a fee), and storing shipments from those stations until the next cycling of the elevator. The station would be zero-g, with all the interesting things that entails. It would require docking facilities for the various transports and shuttles that may be in use, and extensive storage and life support systems as well.

To offset the effects of prolonged exposure to zero-g, a rotating torus could be placed around the cable as well, for use as crew quarters, and as a neutral meeting place for the various megacorps that inhabit the world.

The station and the elevators would be Angelus City property, subject to the government of the city and enforced by City Police Officers. The reasons behind that are primarily political and deal with stuff I'd like to address in another post.

The elevators themselves would best be designed as a tandem system, one going down and one going up, set up like a counterweight in a normal elevator. Various redundant safety systems would be needed, but those can be designed as needed for gameplay. Yes, I'm assuming something will go wrong in someones session down the line.

Ideally, there would be two elevator systems, one for passengers/light cargo and one for heavy construction materials, running at something like a 6 hour trip for people and a 12 hour trip for heavy stuff.