Failover Cluster Manager > Role > SQL Server (Node1 is owner) > Resources (SQL Server & SQL Agent)
My question is, what goes in the connection string for the client
application, the ip for the SQLCluster or the ip for the SQL resource?

You should always refer to the Client Access Name or the IP Addresses you see on the SQL Server role properties.

Also, when I manually failover from Node1 to node2, using the the
roles pane, the owner node changes and the services and shared disks
move over to the node2. But, at the Cluster level, the current Host
Server is still node1.

That happens because one is the management for the Cluster and the other is the management for the Role. A role is related to the service that runs on one node of the cluster and it is somehow independent from the management of the Cluster itself. Therefore, sometimes you can see the owner for the Role and the owner for the Cluster not being the same node at some point.
Sometimes you're gonna see the Quorum disk owned by the node that's not currently the owner of the other disks.