Google Play updates a non-existing application
Why Google Play calls something "an update", if in facts it installs it for the first time?



Even if you think the app was not installed before it may be the case that the app was renamed. Therefore the app name is new, but the app itself was already installed.  



  Disable button for this app is disabled. Can't disable or uninstall it. Any option to get rid of it?



Apps that can't be disabled are system apps. The manufacturer installed it and the app itself is system critical (based on the manufacturers opinion), therefore you can't disable it. With root permissions you may be able to delete the app, however most likely your system will not work correctly at some point (or at all) afterwards.


Regarding the Moto Actions I assume that Motorola uses it for defining tasks that trigger under certain conditions. As those settings are integrated into the OS the app is system critical and can't be deinstalled.


  How can Google Play install anything on my device without asking me for a permission?



The answer is more or less already included in 1. Most likely the app was already installed.



  How can Google Play install anything that I cannot uninstall?



System apps can be updated - you will see that in the app overfiew you have the option to uninstall the update (but not disable the app). In such a case the updated app is installed into the regular app space and the system app is ignored.



  Are there any other app-examples of such behavior?



On nearly every device there is at least one pre-installed manufacturer system app that can be updated via Play Store and that can't be disabled.


If you want a clean Android System you can unlock your device and install LineageOS or another AOSP based custom ROM. Lineage OS for example does not even include the PlayStore and the other Google apps (but you can install them if you want).