Salt is for flavor, of course. If you're a pleb who doesn't use salt, then don't use salt. But you're missing on delicious flavors, herbal salt definitely goes well with egg for me. I have no idea what you mean with your rice and egg question but if you're planning on cracking one open and dumping it into boiling water with the rice, that sounds like a mess. And of course you can cook the rice at the same time as you cook the egg, assuming you got enough space to cook, which you should. You just want to use cooled rice to dump into the eggs so you don't get a messy, sticky meal, which is why the suggestion is to cook some in advance or use leftovers.
Rice gets sticky if you pop it on the pan and start working it when it's still warm. Just check the difference between rice you just cooked and rice that's been in the fridge overnight, you'll get it.
You want your sushi rice to be pretty sticky. You want to use short-grain rice and you do usually let the rice cool for a bit before you start doing anything with it. Dip your hands in water to prevent the rice from sticking to them.