It depends a lot on how sensitive you are to weight while running. The Apple Watch is pretty heavy for a fitness device, but it is not crazy heavy. I bought the sport, because I noticed how much lighter it felt than the SS. My mac daddy Garmin 735XT weighs half of what the AW sport weighs and about 40% of what a SS weighs (with a sport band). Heck, even the piggish 410 that another person referenced weighs 15g less than the AW sport and 25g less than the SS.
I have run hundreds of miles with my AW sport, and it is OK. But, I prefer the lighter GPS fitness watches out there for running. Perception of the weight is all user-dependent, though.
Regarding your other questions about screen durability, this is also highly user-dependent. I have worn watches every day for decades. I have never done any kind of material damage to a screen whatever the screen material. I have had my AW Sport for more than a year, and it is flawless. Similarly, I have had dozens of cell phones without screen protectors, and I think I may have only minorly scratched one screen, and that was a plastic screen. So from my point of view, claims about screen damage resistance are more theoretical than practical, unless you are rough on a watch screen.