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Let’s suppose a space elevator is built along with a space station on one end. It’s tethered to and rotates with the Earth, but is outside the Earth’s atmosphere. What’s the gravity gonna be like on this space station?

(Yes i could research it on my own, but it’s more fun to hear it from someone who knows and wants to talk about it. Feel free to go into more depth about theoretical possibilities that don’t necessarily fall directly under the stipulations of this question.)

The atmosphere has nothing to do with the extent of gravity.

Satellites (including our Moon) remain in orbit around Earth because they are subject to Earth’s gravity. The closer a satellite is to Earth, the faster it must move to balance gravity. Satellites in Low Earth Orbit can go around in 90 minutes. The Moon (much further away) takes a whole month. In between, there is an altitude where satellites will have a period of exactly 24 hours.

Any object freely orbiting Earth will behave as if its whole mass was concentrated at the centre of gravity.

Linking a space station to a space elevator requires that the centre of gravity of the whole system (elevator + space station + counter-mass above the station) has its centre of gravity at just the right altitude for the orbital period to match Earth’s rotation.

If you achieve that, then the entire setup will appear to remain vertical above a spot. If you want the setup to be over the same spot all the time (instead of drifting along a north-south line) pick a spot on the equator.

That is why Arthur C Clarke had picked a mountain in Sri Lanka when he described the principle, decades ago.

Next question is where (along the setup) do you put the space station?

The simple design is to put the station at the centre of gravity (at the altitude of geostationary satellites). In this fashion, the station will be at the correct orbital speed for its altitude. The people in the station will be in free fall (they will fell weightless).

Anyone in the elevator cabin is closer to Earth, therefore will feel pulled towards Earth (but also sideways as the elevator moves up or down, because of the Coriolis effect).

Anyone along the counter-mass, above the station, will be further from Earth and, therefore, turning too fast to remain in a stable orbit. They would feel a "pull" away from Earth (they would stand with their head towards Earth).

If the counter-mass section extends far enough, you could reach a place where the speed of the counter-mass matches escape velocity.

You take your probe in the elevator, past the station and at the tip of the counter-mass side. Release it and the probe is now in a solar orbit, having escaped Earth. Just be careful not to fall outside with the probe.