I am trying to track my Xamarin.Forms application below details using App Center Analytics for enhance our application in the future.
Is this legal? Can I publish my application to PlayStore with App Center Analytics?
Those are (just a few) use cases App Center Analytics was made for.
Tracking custom properties can be acheived through Custom Properties APIs: https://learn.microsoft.com/en-us/appcenter/sdk/analytics/xamarin#custom-events
Or you can use custom properties sdk-wise: https://learn.microsoft.com/en-us/appcenter/sdk/other-apis/xamarin#use-custom-properties
With those, you can track quite a plenty of information you need. Note the limits, though - 200 different events max, and up to 20 properties per event.
As for publishing application on PlayStore it's totally fine, since App Center is basically just an SDK.