How did they measure the distance to the center of our galaxy to 0.3% accuracy?
The orbit of star S2 is completely determined by the astrometric observations. i.e. One has the orbital period, the angular scale of the orbit and the inclination of the orbital plane. With this information alone, the distance and central mass are degenerate. If you know the distance you get the mass and vice versa.

However, here the mass is not needed. Full knowledge of the orbit means that you can predict the radial (line of sight) velocity of the star at any point in its orbit. The conversion from the angular scale of the observations to an absolute velocity, depends (linearly) on the distance to the source.

Hence they fold a set of radial velocity measurements into their analysis and get an estimate of the distance.

The distance is that to Sgr A*. The paper discusses that this is compatible with other measurements of where the dynamical centre of the Galaxy is.

The Galaxy is a proper noun (like the Sun), referring to our Milky Way, and hence is capitalised. When referring to the Galaxy it is conventional (though not strictly necessary I think) to use the word Galactic to indicate a property of the Galaxy (as opposed to some other galaxy).