.NET Core only supports WCF as a client so I'm assuming you're writing your WCF service in .NET Framework.
1) This isn't surprising but ultimately would depend on the serializer being used. Without a setter the value cannot be set which means it would be a calculated value. On the client side that wouldn't matter though. Therefore I would just make the client model allow setting. This would be especially important because serialization in .NET Core isn't the same as .NET Framework. But showing the definition and error(s) would help clarify.
2) Just a guess there is an issue with the TLS protocol and/or cert on the client machine. Ensure the server and client are using the same TLS protocols and that the cert is available on both machines.
3) In the Connected Services UI next to the Add button should be a refresh button. If you press that what happens?