Foodora App logs me out instantly

That’s a safe bet for a start.
Although honestly, with all this installing this and that having been going on now on your phone, I personally would wipe the phone completely and set it up from scratch.

But I know (because I’ve done that a few times myself) that it’s a real hassle and you have to have confidence in your data backup and I understand everybody wanting to avoid that … but it really takes possible pitfalls in the current setup out of the way reliably.

Can’t hurt, but don’t expect quick results.
You could ask them which Google services the App exactly relies on. That would help in picking the right Open GApps package for the moment until they perhaps improve the App.

Yes. If you don’t want to wipe the phone, then remove microG at least, as you can’t count on it to reliably solve your case.
Then install Open GApps.

If all else fails, yes.

Of course you can, but it makes no sense.
Even the smallest Open GApps package, pico, installs the real deal, the Google Play Store.
If you have the Play Store, you don’t need Yalp or the likes.

I don’t have much insight regarding microG because I don’t use it, but it seems reasonable to me that microG and Open GApps could get in the way of each other.
The purpose of microG is to try to look like GApps as much as possible to the OS afterall.