Sigh... I don't know how the internet has convinced people this shit requires some magic incantation to cook eggs. I promise you I can cook eggs in any brand new lodge pan you hand me without issue. In fact if you sand blast the factory seasoning off it first and give it to me shiny I can still do it. And so can you. It requires the exact same skills as cooking with stainless steel which won't take a season no matter how hard you try. Step 1: Bring it up to temp (confirm by tossing some flecks of water in it, if they bead, it's up to temp). Step 2, throw a knob of butter in it and coat the damn pan. Fat is your friend, don't be shy with it. Step 3: Cook the eggs. If a little bit sticks that's fine. You probably should have used more butter - but no worries - just go wash it off in the sink the same way you'd wash anything else. It's a giant piece of iron - you aren't going to hurt it with a little soap, water and elbow grease. When you are done dry it off on the stove and hit it with a little grease/oil/fat to keep it from rusting. If you forget and it rusts... still no big deal - scrub off the rust, give it a little grease and bobs your uncle. This shit is only hard if you decide it is.

Source: Been cooking exclusively on carbon steel, cast iron and stainless for years.

Yes, scrambled eggs works wonderfully in a cast iron skillet but it needs to be well worn in. On a new pan you'll end up with an awful mess. But feel free to go to town on that with washing up liquid, steel wool and a sandblaster if you want. There isn't some magical pixie dust on it you need to worry about rubbing off. Just dry it properly after and oil it before you put it away. Keep frying in it regularly and it'll be fine. A brief hiccup in your skillet's breaking-in process.

Edit: In fact we had scrambled eggs as part of dinner this evening (with rice, chilli crisp, garlic mushrooms and bak choi) and the pan is currently in the sink full of water, where it will remain until tomorrow morning because it's Friday night, dammit, and it will be fine.

Washing cast iron with soap is really fine. Actual seasoning doesn't come off with dish soap.