What happens when Core Cluster Group is failed over to another node? Will it take the whole cluster down? Single point of failure?
When you failover a cluster resource group from a node to different node SQL Server instance and other resources which are part of this resource group is stopped on current node and then ownership is changed to node which you are failing over to. After ownership is changed SQL Server is restarted with owner as new node. During failover SQL Server might go in crash recovery mode so any Inflight transactions( transactions which were running) will be rolled back. SQL Server goes through process of recovery to rollback and rollforward transactions so as to bring database in consistent state as it was just at time when failover happened.

Cluster disk, netwrok name and MSDTC are also part of resource group so they also move to different node.