You have the Instant Support items correct.

Health monitoring simply provides 3 categories of alerts which are viewable on the main HPCM portal page as well as via reports.

The 3 categories are indicated on the UI for HP Health Monitoring task:  Assent changes, Threshold, Hardware health.

Each item listed under each category can be monitored. If a memory module or disk drive has been added/removed, you will get an alert.  If your disk space falls below the specified threshold, you will get an alert.  If your CPU gets too hot or if the chassis has been opened without your permission, you get an alert. 

If you are satisfied with the setting on the UI, set a schedule to roll out these settings to your clients.  Once the client gets the new settings, the HPCM agent on the client will start to monitor for these settings and present local alerts as well as send the same alerts to the HPCM console where you can view the alerts from the HPCM portal page as well as the vairous reports.

How do these interface with the local alerts setting. I had assumed that these three categories tell the server about problems while the local alerts enable/disable control which items the user sees. However, in another topic you said you could not disable local alerting without disabling alerting itself, so now I am unsure.

EDIT: In addition are there any plans to add times and dates to the portal warnings? I can see that PC X had a disk warning, but it seems to stay there long after its been resolved and so the warnings become useless.