Another cast iron fan here. I've been cooking with it since I was a kid (and still feel a little guilty about the time I scrubbed and scrubbed and scrubbed my mom's skillet to get rid of all the black stuff and make it shiny for her. I didn't know...)
I use it for pretty much everything except boiling water (pasta and hard boiled eggs), and candy making (would probably work, but don't have one the right size).
As long as you have a good seasoning, you can do whatever with it. Use dish detergent, or don't, doesn't matter. I do the simmer-on-the-stove method to clean out stuff like sauces, but even when I soaked I never "ruined" my pans. Dry it on a burner set to low, or in a low oven, to avoid rust (and, if you do get rust, you'll need to reseason it, because you have bare spots).
No, I can't throw it in the dishwasher like I can my Viking (made in USA) stainless steel pans, but I really don't see how they take any time consuming extra care or maintenance. Maybe because it is all just automatic at this point.
As for eggs sticking-- may not be seasoned well enough (you want lots of thin layers to start), or you may be cooking at too high a heat. Try a lower heat, a little oil, and don't turn them too early. Stuff tends to release from cast iron on its own when it is ready to turn. With a really well seasoned skillet, you can skip the oil. I usually let the pan start heating up while I get things out of the fridge, etc.
