Thanks for your help Peter and Sandeep! :)
For some reason, the Eclipse timer now does not time out and the server status becomes Started. I don't know why it started working right. I did delete some servers from the workspace and the filesystem and reinstalled the servers. That probably had something to do with the servers now starting.
I am still having problems, and I am trying to understand the problems and characterize them, but at least I am moving forward.
I have several servers installed on the Eclipse workspace. Maybe I have not correctly set all of the ports. Maybe there are some conflicts between the various servers.
Thanks again for your help!!