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