alexandros Sorry for not responding sooner.
You can download and install apps outside of an app store, for example you can download a lot of apks off of GitHub. But it is generally safer and more secure to use an app store.
Logging in with a specific account would tie that IP address with the account, you could use a VPN to mask the IP address. Even when using something like Aurora store, my understanding is it is still connecting to Google servers to download the apps, so there is still some information like IP that would get logged.
You could theoretically create multiple throwaway google accounts and cycle through them, although I am not sure if it actually gets you much more privacy. I guess to some extent it depends on what your privacy model is like and what you are trying to keep private.
If you wanted to do the absolute minimal contact with the Play Store, you could do something along the lines of having a dedicated user profile on a VPN that downloads the Apps from the Play Store. Use a computer or USB drive, export the APKs and then go install them on your normal user profile(s). Then when you want to update an app, log into the "Google Play" profile, update the apps which will update them on all the user profiles.
Not sure that answers the questions, but hopefully is helpful.