1. Is material on Earth's surface not in free fall around Earth's center?
No. Material on the Earth's surface -- or inside it -- is not in orbit, and so is not in free fall. You can temporarily put yourself into an orbit (and thus into free fall) by jumping up into the air, or jumping off a higher surface. When you do this, you are briefly in a very eccentric orbit (one which would take you very close to the center of the Earth, if the Earth wasn't a solid body) -- but then you hit the ground and are no longer in orbit.
The Earth rotates in the same way that a spinning top rotates; this has nothing to do with orbits.
2. How are geostationary orbits even a thing? Seems like the only orbit that could be geostationary would be standing on the Earth's surface.
Again, the surface of the Earth is not orbiting. The Earth rotates as a rigid body, with (as AtmosphericPrisonEscape noted) residual angular momentum left over from its formation, like a spinning top.
Because your angular speed in an orbit decreases the further away you are from the Earth, there will be a point where it happens to match the Earth's spin rate. If you arrange the orbit so it is above the equator and in the same direction as the Earth's spin, then you will always be above the point on the equator: a geostationary orbit.
3. What changes as you orbit further above the Earth's surface? Does your angular speed increase or decrease? Does your tangential speed increase or decrease?
Both your angular speed and your tangential speed decrease the further away you get. (Your angular speed would decrease even if your tangential speed stayed the same, because the circumference of your orbit increases with altitude; but in fact the tangential speed decreases as well.)
4. Is magma near the center of the Earth not rotating faster than material in the crust, like in an accretion disk?
The Earth rotates approximately as a rigid body, so in general, no. The molten outer core (which is not magma) may rotate slightly slower, while the solid inner core might rotate a little faster, but we're talking about $\sim 0.1$ degrees per year differences, and this has nothing to do with orbits. (The Earth is nothing like an accretion disk.)
5. Can two objects be orbiting (circularly) at the same altitude but with different tangential speeds?
Ignoring minor deviations due to things like the non-spherical nature of the Earth, mass concentrations in the crust, etc., the orbital speed for a circular orbit is a function of the altitude only. So two objects in circular orbit at the same altitude must have the same tangential speed. (Note that they can have different velocities, because velocity is a vector quantity -- so you can have two object orbiting in different -- even opposite -- directions at the same altitude, at least until they run into each other.)