On our summary page we have an Active Applicaiton Alert section as seen below.
If I have applications that are currently in critical status as seen from the summary below, shouldn't they show up in the Active Applications Alerts as well?
Have you set up any alerts to get triggered when an application's status changes? If you haven't, you won't have any active alerts to be shown in that resource.
Have a look at Configuring Orion APM Alerts to see an example of creating an alert that writes to the Orion event log if an app goes critical.
You can of course modify it to send e-mail alerts, page people through SMS, and so on. More information on that is in Creating and Managing Alerts.
The resource that tells you exactly which of your applications are having problems is the "Applications with Problems" resource.
Is that resource missing from your summary view? I can show you how to add it back.
I have the alerts set up properly based on the 'APM Alerts' link you provided. What I don't see though is the Application with Problems when I try to add a resource in the customize page section.
APM v3.1 added a customized "Applications by Status" resource named "Applications with Problems". To set it up if you don't have this, add an "Applications by Status" resource to the summary page, and then Edit it:
To get back to your original question, if you have set up alerts that you know are in a triggered state but are not appearing in the active alerts resource, I really think that warrants a call to support. It certainly isn't typical behavior.