Incidentally I talked with Esri Support and thought I figured out this issue.
For others reading, if you want to use SSL and ArcGIS Server and the web adapter on the same machine, do not setup your CA Cert on arcgis server. Use a self-signed certificate on ArcGIS Server. The Web Adapter (with IIS cert) will act as a reverse proxy and will your requests will appear to the outside world correctly. However, this means that you cannot use 6433 for administrative purposes when accessed from the outside.
To answer your quest Jeff:
* Yes I have restarted the ArcGIS Server service
* Now the arcgis server locks up after a few hours. Although the folders in the rest end point still appear, all of the services are gone. This is pretty strange, because you would figure if arcgis server is down than nothing would appear.
* The previous error still appears in my logs, however now it is followed with some new fun:
<Msg time="2012-10-30T18:40:35,14" type="WARNING" code="7709" source="Admin" process="4312" thread="1" methodName="" machine="LOCALHOST" user="" elapsed="">The App Server was found to be stopped when it should have been started. Restarting it.</Msg>
<Msg time="2012-10-30T18:41:38,174" type="SEVERE" code="6749" source="Admin" process="4312" thread="1" methodName="" machine="LOCALHOST" user="" elapsed="">Error while starting AppServer. Required port '4,000' already in use.</Msg>
<Msg time="2012-10-30T18:41:38,174" type="SEVERE" code="6550" source="Admin" process="4312" thread="1" methodName="" machine="LOCALHOST" user="" elapsed="">Failed to start the server machine 'LOCALHOST'. Error while starting AppServer. Required port '4,000' already in use.</Msg>
To me this looks like it decided TomCat is not running, then decided to restart it. However, TomCat was actually running and now ArcGIS server is totally confused at what to do next.
I need to do some more diagnostics to fully understand what is causing this.