I have solved the problem of not being able to send e-mail from my BlackBerry and getting the e-mails at my home e-mail address. I will explain the problem in case anyone else suffers from the same problem.
I use my BlackBerry for e-mail with my company BES. My company computer is a notebook, and since I travel all the time, it is not connected to the company domain. I sync my BlackBerry for everything but e-mail with my home computer, for Address, Calendar, Memos, and Tasks. I have the BlackBerry Desktop manager installed on my home computer where I have a personal e-mail address, different from my company e-mail address that is used with the BES.
When I installed the BlackBerry Desktop Manager, one of the questions is are you going to use it with a BES. I answered yes, and the result of that answer is that the Redirector is installed as part of the BlackBerry Desktop Manager. And having Redirector installed on my home computer was the problem. Redirector is designed to be installed on the same computer that works with the BES. It turns out that the e-mails I was receiving that I described above were being generated by Redirector.
So the solution was to uninstall BlackBerry Desktop Manager, reinstall it, but tell a little white lie, that I don't use it with the BES. When you install it that way, Redirector is not installed, and after wiping the Blackberry, reactivating it with the BES, and then syncing it with my home computer for calendar, address, memo, and task, everything works fine.
Part of my problem was that after I started experiencing the 3 problems listed above, I called Verizon. Their technical support person told me I had to have Redirector installed to use the BlackBerry with the BES, a wrong answer. The same person also told me that being able to search after pushing the phone key was taken out of the newer version of the software. Both answers were clearly wrong. It is sad that Verizon can't provide techinical support people who know the answers.
I am left with one small problem and that is that my bluetooth headset does not automacally connect when I make a call. I have to push the button on the headset after making the call. It used to automacally connect before the upgrade, and I still don't understand why it won't now.
Anyway, thanks for those of you who helped me get through these problems.