I've made them with the 'boil, cover, and sit' method for years with various stoves (all electric though I think) and it's always worked. The electric stove does stay warm after you turn off the heat, maybe that's why? I've *never* had runny eggs. I have noticed that if you take them off the second they boil, the yolks will be cooked, but tender (which is how I like them.) If you leave them boiling for a while (which, yes, often happens when I'm on MDC, so I can relate
) they cook *hard* and aren't as good, IMO. They also tend to crack and leak -- and you get little bits of poached egg instead.
I always thought the green rim was caused by refrigerating them? I never notice it just after cooking. But maybe that's just because, like a PP conjectured, it's because I always rinse with cold water. I'll have to let them cool naturally and see what happens.
Oh, and a tip -- use "old" eggs. That is, eggs that have been sitting in the refrigerator for a couple of weeks (assuming they were fresh when you got them.) There is some protein or something that attaches the white part to the shell, and that breaks down over time, making them easy to peel.