Disclaimer: This is not my field of expertise

Okay I will try going through this piece by piece:

• has there been or could there be a time where someone actively looking at the sky sees a new star appear before their eyes - The probability to experience a new star with naked eyes or even a regular telescope is very low. You will have to know the exact location where you would like to look into the sky and perhaps the timing too. - As @Bjorn says we have observatories working 24/7 around the globe and new stars are always popping up.
• does it have a limited range before it would just always be invisible beyond a certain range based on size and luminosity, like a flashlight? - Photons can travel the vacuum forever. Flashlight has limited range only due to photons getting diffused or barricaded by some other molecules present in their path.
  - However, we know space is not exactly pure vacuum and that means signals we receive from the stars are still going to be inaccurate.

Since our discussion was mostly on eyes, I found this article for you, hopefully this satisfies your curiosity. :) https://www.johncbarentine.com/astronomical-visibility.html

Yes, it has already happened. When you look.up the sky, you are pretty much looking at ghosts. Due to the speed of light being finite.

Honestly, this is too good of a news to be true. There has to be some kind of error somewhere. Not used to this.