I'm afraid that the "design" part will always evade me. I'm better at the plumbing that goes behind the walls. For my next adventure, I'm hiring a graphic artists/web designer who's previous work has impressed me.
It's not really the same, because everyone is exposed to design all the time, whereas only coders ever look at code. It's more like being able to tell if music is good -- most people have a decent sense for it, but a musician's taste is a little more refined.
Well, I can recognize that the guy I'm working with for design is great, even though I can't design the way he can. What he does really impresses not only me, but others who see his work, so I do have feedback besides my own.
No, in the "art" worlds much of what is respected by other artists is completely incomprehensible to ordinary people. Unless you're trying to win design competitions, the measure of a good web design is how well it is received by your site's typical users.