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 object we are studying might not move in the same repeatable orbitable pattern around our (0,0)... like say a comet or asteroid that is moving in a straight line out of any given orbit.

Comet and asteroid orbits are just the usual conic sections, most often ellipses. Straight lines would require propulsion; actually, a bona fide "space drive". Once the orbital parameters have been found out, the orbit can be extrapolated for tens, even hundreds of years with some accuracy, and assuming there are no close encounters with massive objects.

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.

The celestial coordinates are the same for everyone, by definition. But an individual observer must account for when and where on the Earth they are looking from to know where that point will be on the sky.

Astronomers use sidereal time, which represents one revolution of the earth, roughly 23 h 56 min. No, a day is not 24 hours long. So we use coordinated AND siderial time.

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).

Also, FYI the North Star isn't truly fixed either as our rotation axis is processing. The Earth is more accurately like a spinning top. It's a slow change, but in a few 1000 years there won't be a North Star. Or at least it might be a different Star, not Polaris

they do. Earth's north pole projected onto the sky is the North Celestial Pole, and Earth's equator is projected onto the sky to give the Celestial Equator. Where the Sun's path in the sky intersects the Celestial Equator in spring defines the zero point.