I don't think Aurora can provide that, can they?
You can probably download the apps you need with Aurora Store. There are some apps that refuse to work unless they were installed through the Play Store. You would have to try to see if your apps have that requirement or not. Although, Aurora Store does not automatically update apps and isn't very reliable.
how on Earth can I avoid giving away a phone number?
This has been discussed several times in other threads and I'm not going to say anything that hasn't already been said, but I'm gonna say it anyway... Either you create an account from a normie-looking network like your home wi-fi or public wi-fi or you get lucky with a VPN.
What I did was to create a Google account with VPN and gave a temp phone number for SMS verification by using 5sim.net. It cost something like 15 cents to receive one SMS with a random phone number. There are more services like this.
what if they (google) will ask me periodically to verify my account...
Google only needs to do initial verification. They don't pester with other verification requests later (as long as you don't do anything weird) so it's safe to delete the number from the Google account afterwards in my experience. It continues working.
But indeed, untoggling some permissions may cause the package to malfunction so what do I do?
If you only need sandboxed Google Play to install/update apps and/or to handle notifications for messaging apps, you can revoke all its permissions aside from Network. The Sensors permissions can be revoked. Just because an app tries to access Sensors doesn't mean it needs it to function. If you don't want the Sensors permission to be enabled by default for newly installed apps, toggle off the switch in Settings > Privacy > Allow Sensors permission to apps by default