Every object in the sky has precise absolute coordinates. As far as I know, there is not a comprehensive database of all sky objects but several databases from different sky surveys that map only a part of the sky. To check if the object is new, you just look up the coordinates in a program that aggregates all the different databases, like Simbad (https://simbad.cds.unistra.fr)

Does the current location of the earth affect that coordinate system at all, or i is negligible due to distances?

They use a Galactic Coordinate System, which is actually centered on the Sun, in the plane of the galaxy. (The sun does revolve around the galaxy, but with a period of millions of years it doesn't make a measurable difference.)

https://en.m.wikipedia.org/wiki/Galactic_coordinate_system

It can easily be transformed into telescope pointing directions with the input of location on the planet, time of year and time of day.

IIRC the Earth has made 2.5 orbits about the galactic center since its [the Earth's] formation.

The period is ~250 million years. Quite the journey! :)

Up to ~300 light years away this effect can be used to estimate the distance to a star.

For other galaxies the effect is very much indistinguishable from 0.

https://lco.global/spacebook/distance/parallax-and-distance-...