General relativity follows essentially from two postulates, the general principle of relativity (which says that physics does not depend on the choice of coordinates) and the equivalence principle (you can’t locally tell the difference between acceleration and the effect of a gravitational field). Both are, of course, not proven – they’re postulates, and the other forces depend on postulates of their own – but it would just be a weird universe in which it made a physical difference what numbers we use to coordinatize a manifold, for example. And the understanding of gravity that follows from these assumptions is just as complete (or incomplete) as we have for the other forces, and just as much in account with experiment.

He’s talking about grand unified theories (GUTs). In those, all the microscopic forces follow from a unified principle, but they are ‘broken’ apart into three separate forces in the current state of the universe.

Just a minor point, but in GR, mass is not the (sole) source of gravity, stress-energy is. Stress-energy is a ‘composite’ quantity, whose components are, roughly, the energy density, the energy flux, momentum density and flux, pressure, and shear stress (‘mass’ just being a kind of energy here). So you’d just need to produce the appropriate stress-energy.

Well, we know there’s something that behaves very much like anti-gravity, known as dark energy. This can be, for instance, sourced by a fluid with negative pressure, and there’s no a priori reason such a thing can’t exist.

Spinning your ship is gravity in the same sense as mass is; it leads to a certain stress-energy, which produces a certain space-time curvature. As an example, take the Schwarzschild metric, which describes a spherically symmetric spacetime at the center of which sits a massive object. Spin that object, and you get the Kerr metric, very different from Schwarzschild’s.

It’s gravity, in the sense that it’s locally described the same way as if you were in a gravitational field. Note the locally here: if you go to any finite region, you will be able to perceive tidal effects, which are absent in the case of linear acceleration. However, if you stipulate an infinite massive plane, you’ll also have no tidal forces.