I'm no expert here, but I'll try to guide you to the best of my knowledge, as we have a Kids app and we are in similar situations to yourself.
1) Can I publish my app, that is not present yet in the app store, outside of the 'kids category' for now (and keep my 3rd party analytics)? (so my app is at least live / installable for my users)
Technically, yes you can, BUT when you change the app to KIDS category, then you can NEVER switch back.
More on this below.
2) If yes to question 1: How would I go about moving my app outside of the kids category? I can't find a place in the UI of the app store connect site to change it.
Not sure here but I understand it's possible. I think it has more to do with the app rating than anything else.
3) If yes to question 1: Can I, at a later time, change my app to make it conform to the 'kids category' requirements and then move it to the kids category?
I think so. The thing here is that if an app "looks" like it's a KIDS app, in apple's of COPPA's eyes (i.e. it has "kids" in the app name title, or has art that looks that the app is for kids) then apple automatically asumes it's for Kids, same thing happens with Google Play.
That means that the only way to achieve this would be to upload an app that LOOKS for adults, and then change all the assets and titles to a kids app... Anyway, apple review won't like this one bit.
4) It is unclear to me what Apple means by 'third party analytics'. Is the sending of in-app events with the device id to my own backend acceptable (is this considered 1st party analytics)? Or is Apple referring to it's own analytics?
Third party analytics: firebase, unity analytics.
If you have your own analytics thing, you're ok.
5) If it is unacceptable to send analytics to your own server, how can we log in-app events? The analytics provided by Apple does not seem to include a solution for logging custom in-app events.
That's acceptable. COPPA allows you to use, even, third party analytics. Apple allows you as well, but only if you DO NOT share any info with Third Parties. So, that's a gray area, because the definition of Third Party for COPPA is someone OUTSIDE your company and OUTSIDE the tools you use: that means that, Firebase, or Unity Analytics, in the eyes of COPPA, is NOT a third-party.
BUT, lately in the eyes of Apple, any company that is not your own (Unity analyics, Firebase), IS a third party.
We're struggling with this issue ourselves, trying to launch the app with Firebase in it, but Apple Review is not accepting it, mostly because it uses IDFA. But even if we removed IDFA they are still requesting us to remove Firebase.
This makes no sense, really, because Firebase (Google) does NOT have access to our info, just ourselves (it is used plainly as a tool?, and no other entity has access to it, so, we're scratching our heads on this one right now.
6) Do I need to remove all microsoft app center code from my binary even though i'm not using it in my code? If yes: I have a xamarin app with a shared codebase, so how would I go about doing that?
If you have unused code in your app, Apple will request you to remove it. You can try though.
7) The section uses 'may not' and 'should not'. Since I am not a native english speaker I am wondering if these mean these same thing in this context.
May not and should not is the same.
Hope we can find a solution to this issues together!
