Are you saying every spacetime has this property? If so, that's not correct; many spacetimes have no isometries.A spacetime can be foliated by spacelike hypersurfaces which are related by an isometry ##\partial_t##.
Since this is an advanced source, it is assuming that readers already understand in detail the procedure that is being described here; so it doesn't have to explain that procedure in detail and with rigor. You should not be taking this wording to be part of such a detailed rigorous description.Look at the sentence below the caption of figure 4 on page 8 of the notes.
integral curves of this vector fields
a timelike congruence
the timelike vector field that generates the isometry
Yes, that is what "static" means: that there is a timelike Killing vector field (isometry) which is hypersurface orthogonal (the spacetime can be foliated by a family of spacelike hypersurfaces that are everywhere orthogonal to the integral curves of the isometry).Isn't the isometry there because you are talking about a static solution for the metric in both cases?
The classic GR textbook that goes into this in some detail is Wald (1984). The more advanced classic source is Hawking & Ellis (1973).Could you point me towards the references where I will be able to learn the following with rigor.