Inert, when talking of cooking ware, typically means "non-reactive"; it won't leech the metals of the vessel into acidic foods. Copper, cast iron and high-carbon steel are all "reactive", where annodized aluminum, stainless steel and enameled cast iron are non-reactive.
The non-stick properties of enameled cookware vary from manufacturer to manufacturer, but in general, they are slicker than stainless or anodized aluminum, but less or equal to Teflon-coated pans. Higher quality, thicker enamel coats tend to be more slick than the cheaper quality stuff. Lodge tends to make some very good, quality-consistent stuff.
Ceramic pans are as inert as cookware can get - even more so than steel. I bought a large ceramic GreenPan skillet a few months ago as part of getting rid of all plastic in my food-related pans and things. It's plenty of €€€, weighs a ton, and has mixed reviews on Amazon, but it works wonderfully for me.
I use it a few times a week, wash with soap. It's not as easy to use as a teflon pan, as it requires seasoning (read: oiling) when cooking eggs and meat. But it cleans easily even if you burn food, and as long as you don't use anything abrasive to clean it it should work fine for a good while.
I cook my eggs in a ceramic pan every morning. I like it better than teflon and it's easy to clean. I wouldn't worry about leaching very much, ceramics are going to have heavy metals in them, especially chromium, but they're all bound up pretty strongly in the ceramic, so even if anything does come off into your food it'll be fairly non-reactive.
I have a small one I cook eggs in every morning. I also use it to heat up leftovers. It works like my Lodge coated dutch oven. It's really slippery. So far it has lasted longer than the "eco" non-stick pan it replaced that cost about $30 more.
I have a set of ScanPan ceramic-coated, non-stick cookware, and I love it. It's quite expensive (about 500-700 for a set these days, depending on the set and where you buy it), but it comes with a lifetime warranty. I've needed that a few times, since the coating has chipped on two different pieces over the past 7 years. Also, I need to wipe down the pan/pot interiors with a paper towel after hand-washing (although the manuals says they're dishwasher-safe), because grease does sticks to them otherwise.
Regarding eggs, I use enough bacon grease that the eggs float on it, so that's never been a problem.