Yes, a full coordinate system specifies not only the location, but also the time of the equinox. This way, the location at any other time can be calculated. You can read about the precession of the equinoxes to learn more about this.

The zero latitude line (usually known as zero declination) is an imaginary line across the sky that is directly above the Earth’s equator; it’s known as the celestial equator. Since that defines the zero declination (in the same way the Earth’s equator defines zero latitude), then the next thing astronomers needed to do was to choose a point along that line to define (0,0). Rather than choose an arbitrary star, which may later prove to not lie on the celestial equator with more precise measurement, they instead chose the point where the Sun’s apparent path across the sky (known as the ecliptic) crossed the celestial equator. Two non-parallel lines must cross at a point, so this made it a good choice. I believe the choice of adopting the crossing point associated with vernal equinox was arbitrary - they could have equally likely chosen the autumnal equinox.

You are correct that the exact location of where the celestial equator and ecliptic cross will change slowly over time, so now astronomers have defined precisely where (0,0) lies relative to reference celestial objects that do not move over millenia. This point is, however, almost exactly at the traditional location defined by the celestial equator and the ecliptic.

The plane in which the Sun moves in the sky and the plane of Earth's equator intersect on a line that points on one end at the zero-zero point.