The North Star isn’t on the celestial equator, so it doesn’t have a zero declination. It’s also not exactly aligned with the poles (it’s just under 1 degree off, which is a large offset for precision astronomy).

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.

We use something called ephimerides, which describe how much an object moves in longitude and latitude (Right Ascension and Declination) per day etc. In other words, they describe the trajectory of a moving celestial object. The vast majority of celestial objects don’t move appreciably on the sky from one year to the next, the exceptions being solar system objects (planets, comets, asteroids) and a handful of nearby stars.

The sky has a similar set of coordinates as the Earth - in fact, the coordinates have the same names: latitude and longitudes. You’re correct in that the sky appears to move above us as the Earth rotates, but we astronomers have solved this by specifying a point on the sky that we define as always having coordinates (0,0). So, from the point of view of someone on Earth, the (0,0) point moves as the sky apparently moves above us.

The choice of where we’ve decided (0,0) to be is somewhat arbitrary, but if you’re curious, it’s defined as the position of the Sun on the sky at the vernal equinox (around March 21st).