A question on gauge fixing

Yes, all those things are correct.

The equivalence class of potentials that are related by gauge transformation is called a gauge orbit, since it is an orbit for the action of the group of gauge transformations on the space of potentials.

Choosing/fixing a gauge means picking out particular representants $A$ from each gauge orbit according to a rule encoded by $F[A] = 0$ for some functional $F$, i.e. you select those potentials which fulfill the equation. A partial gauge fixing is indeed given by things like $F[A] = \partial_\mu A^\mu$, for which $F[A] = 0$ has more than one solution in a given orbit. These solutions are related by gauge transformations with harmonic parameter functions, and these transformations are called residual gauge symmetry.

In general, it is not possible to fix a gauge that choose only one representant from every orbit. This is known as the Gribov problem.