Quote by peter46464
1. The problem with a Newtonian inertial frame in GR is that (because of the equivalence principle) you don't know whether it's in a gravitational field or if the frame is accelerating). That's why the definition of an inertial frame in GTR is one that is freely falling. Is that more or less correct?

Yes. It is inertial to first order in spacetime derivatives on the freefalling worldline, so local means, in part, at a point. Even at this point, it is not inertial at second order. The technical implementation of this is called "Fermi normal coordinates". The restriction to first order at a point is the technical implementation of "local" in the statement of the equivalence principle.
Quote by peter46464
2. Is it possible to have useful approximations of an inertial frame in STR with a weak gravitational field? I haven't phrased that very well. I think I mean is STR still useful on Earth (for example) even though there is a gravitational field on Earth?

Yes. For points near a freefalling wordline, Fermi normal coordinates shows in an order by order expansion how they deviate from inertiality. For points near the freefalling worldline, the deviations may be so small as to be negligible. In practice, "near the worldline" is large enough to include entire particle accelerators, which only take SR into account, not GR (unless they use GPS, like OPERA
).