Should health monitor be the port on the node?

You are right; monitors should verify the health of the service you're delivering. Generally, it is best to configure monitors to verify representative requests. For instance, for HTTP responses you will want to look for an actual "200 OK" response rather than simply verifying that a connection can be made on port 80. However, if a "200 OK" doesn't necessarily mean that the app is OK you'll need to be more specific (and look for a string, for instance).

Monitors can be assigned to nodes, or to pool members. They don't monitor pools, just pool members.

There are several options to be notified when a service is unavailable. For instance, you could use an SMNP-trap, or use an iRule to generate a message. You could also use an external monitoring/notification system (e.g. Nagios, or a cloud-based monitoring service).