I love this stuff! I can't wait to get my chicks grown and laying to try a few of my own experiments. Kudos to all of you on testing and sharing your experiences.

On appearance of eggs:
I did some research on local egg producers and sales before deciding what breeds of hens to get, since I figured I'd be selling my own excess eggs at some point. I hung out at the farmers markets and local feed/produce stores, made some friends, asked questions, watched and learned. Around my neck of the woods, the multicolored dozens of eggs always sell on sight to the local young professionals and the hipsters. The deeply colored yolks keep them coming back to a particular source, as they equate those deeply colored yolks with wholesomeness and "richer tasting". The older ladies who bake and cook generally want the freshest and tastiest eggs, regardless of the color of shell or yolk. No matter the age group, it seemed that people who knew nothing of chickens or eggs thought the colored eggs (including brown, green, blue, etc) were all "organic" and "healthier" than white eggs, though a very few thought they had been dyed. Even my HE thinks brown eggs taste better than white ones, an idea I sometimes think I've disabused him of only to have it reappear.

Folks in your area might be different. If you aren't selling your eggs, just do what makes you happy.
On nutritional content of eggs:
I do believe feeding a nutritious diet to your hens gives you the healthiest eggs. I've seen studies that show increased omega-3 in eggs from hens fed with feed fortified with increased omega-3. No mention that I recall in that study of yolk color, though I do believe palatabilty was stated to be either unchanged or of negligible change.
Beta-carotine is an antioxidant, a precursor to vitamin A, and lacking in diets low in vegetables. Most Americans don't eat adequate amounts of vegetables high in beta-carotene or vitamin A. Having a bit extra in my eggs couldn't hurt, as the amounts one would have to consume to become toxic are unreasonable.
On flavor of eggs:
The only time in the past that I remember our eggs tasting funky was after killing a skunk right next to the henhouse. The skunk stench made the eggs smell and taste awful for a couple of weeks, if I recall correctly. Didn't do much for the smell of the henhouse either. We did try washing it, but . . . skunk. My mother says the eggs from her father's turkey farm tasted bad when the turkeys ate stink bugs that were a plague one year. I read somewhere, many years after hearing this family story over and over from my cradle, that barnyard fowl wouldn't eat stink bugs. I don't know. My hens years ago ate all our garden and plant-based kitchen scraps (meat and dairy went to pigs), including onions, hot peppers, and garlic. I never noticed any effect on their eggs or flesh. So as far as changing the flavor of your eggs, I can only guess that a more "robustly eggy" flavor might be produced by well nourished hens as opposed to malnourished ones, or some breeds might produce a slightly different flavor, but I don't know what exactly you'd need to feed to get a major difference beyond that. I'm interested in learning if it can be done consistently, and how.