If my monitors native res is 1920x1080 and I run something lower..
 

Does that affect the quality of full screen HD videos I watch? I keep thinking it's not, then I think it is.. I can't decide. I think it probably should be affecting the quality.. right?

Ya, you don't want to do that if you can avoid it.

Anything less than the native resolution is less than best quality.

Anything other than native at all.. The days of adjusting your screen are long gone, you need to run what the manufacturer says or it will always look funky.