I have a 75inch smart TV and a macbook air. I would like to watch a movie on DVD from the library. Since I need to get an external drive to watch it and then use the HDMI to put it on the TV, do I lose quality by not having it on my original system?
No. The MacBook renders the video either way. Any hard drive should be able to supply more than enough bandwidth for a single video stream.
I think the external drive the OP is referring to is a DVD burner not a hard drive but ...the answer is the same - no lose of quality with an external vrs internal.
Thinking of external DVD units - for not too much more you could get an external Blu-Ray reader - given the size of your tv - high definition would be nice. I know you specifically mention dvds from the library but more and more they are getting blu rays.