Karen,
Part of the problem description that you gave sounds similar to these threads.
https://community.esri.com/message/389700?sr=search&searchId=0d4b19f7-f6f6-4850-999c-d87daebc8e92&se...
https://community.esri.com/message/360412?sr=search&searchId=c6ae554a-37c8-4fe4-a8ac-0ab72ac6d51f&se...
Adjusting your Internet Explorer LAN Settings may help.
For the hang on start issue. If ArcGIS Explorer starts some times and hangs other times, it may be an internet connection or service access problem during the start up. If ArcGIS Explorer does not get a response from some of the service requests, it doesn't time out, and just waits (hangs).
This can be tested for by disconnecting your computer from the internet. On a computer where you have successfully run ArcGIS Explorer in the past, if you disconnect from the internet, ArcGIS Explorer should still start up, a '502' response will be returned and ArcGIS Explore will just move on with the start up.
If while disconnected you still get a hang, then we would want to look at things that may have changed on the computer.
The #1 thing to look for is a system update the changed a graphics card driver.
Was anything (other software) done that changed graphics card settings?
If you have a default map "My Default Map.nmf" in your Documents\ArcGIS Explorer folder. Rename the file so ArcGIS Explorer will not open it on start, perhaps there is something recently added to it, that is causing an issue ??
Mark