Apple Watch WiFi

Are you using Wi-fi calling? For Messages, are you using iMessage or SMS? If this is a GPS only Watch, only certain features are available when the paired iPhone is out of range. See this, Use your Apple Watch without your iPhone nearby - Apple Support Make sure that you read the footnotes carefully.

Okay, that can address some of the iMessage issues. You did not say if you have wi-fi calling activated with your carrier so you could be doing phone calls on the Watch.