Availability Group Clustering - Cluster Group Preferred Owner
Yes, you need to add a DR site IP address for the cluster name.  It may not be critical if you aren't using the cluster name for anything, but it is certainly a best practice and considered the correct configuration.

You don't really need to be concerned about the core cluster resources (cluster name, fileshare witness, etc.) failing over to the DR site.  The cluster will run just fine if that occurs, and if the network to the DR site is interrupted, the two nodes in the local data center will arbitrate for ownership of the core resources and one will bring them online.  This will not cause the availability groups to go offline and come back online--that will only occur if the AG primary can't retain ownership of the AG resources.

Having said that, you can set the possible owners of the cluster name resource on the Advanced tab, but in the case where you need to fail over to the DR node, you would need to change it again in order for the resources to come online on the DR node.