"...set up both the phone and Verizon.net email. Last night I received dozens of email. Some messages dating back to 12/30 that were previously deleted from my home PC. Some messagesI received 4 and 5 times, as they continued to sync to the phone after deleting them from the phone. I am not receiving any messages today. I got the “unable to open connection to the server” message once and the remainder of the time get the perpetually spinning circle with the “loading messages” statement. ..."
I'm sorry that your experience with the Droid has been so unpleasant. Mine has been just the opposite. Sure there have been a few quirks, but nothing that I wouldn't expect to see on pretty much any platform in one form or another.
I don't have a verizon.net account so I don't know that I can be much help, but the underlined text above makes me think that this may be more of a problem with verizon.net than Verizon Wireless or the Droid itself. If you had previously deleted the emails from your email account (not just within Outlook or whatever email client you are using) then it's pretty much impossible for the phone to download something that doesn't exist. From that I would assume that you have your verizon.net account setup to "Leave messages on server". So it's only natural that when you set up the account on the phone that it will sync ALL the messages on the server, not what's showing up on your PC's email client.
Also, for the italics text above. If you did not allow the sync to finish before starting to delete messages on the phone, it's possible that it was re-syncing them because of that. Or, maybe there is an issue with deleting them before marking them read.
The "unable to open connection..." problem may be a problem on the server end. Maybe not. I'd try a battery pull / reset and see if that fixed the problem.
If you are retrieving your emails from your verizon.net account from Outlook, I'm sure you have Outlook set to "Leave messages on the server" for a certain period of time... (It can be found in the Advanced settings menu of the account in Outlook - OR outlook express)
There is an option for the "number of days" to leave them on the server, since your phone will likely be checking hourly, or at least daily, set that option to only 1 or 2 days. There is also an option there to "Delete from server when emptied from recycle bin" (or trash, or what ever they call it...)
If you check that box, and remember to empty your trash folder before exiting, and do a send/receive just before you close up, those messages will be removed from the Verizon server.
Outlook doesn't always "play nice" with email servers, and sometimes messes with the "last message retrieved" settings on the mail server itself, which could possibly be causing your problems. Many POP3 servers have settings to handle this "broken" behavior of Microsoft's products, but that is out of your control, unfortunately, as it would be on Verizon's side at the server level.
I've found, with some of my clients who have issues similar to yours on other devices, that setting the "retention period" to a very short time like 1 or 2 days helps with these issues. Also, when it *does* happen, usually going into your Outlook settings, removing the "Keep messages on server" check box, so that it downloads and deletes everything from the server, performing 2 Send/Receive cycles, then going back in and turning back on the retention will usually "reset" all the counters on most mail servers, so that your mobile device will start to sync properly again. (Of course, any messages not on the device at the time you turn off the retention in Outlook and do the 2 syncs, will never come through to the mobile, but you can look at those messages and forward them to yourself if you need them on the mobile.)
Consider making a change to K-9 Email. I was having numerous problems opening attachments, getting emails (the revolving update symbol that never went away), especially when I tried to sync to my email account that I leave all messages on the server and it has literally thousands of emails going back a couple years. K-9 has no issues with that account. I have struggled to get K-9 to sync with a Microsoft Exchange email account, which the native Droid application seems to do just fine. So, I have three email clients on my Droid (because I also have some MS Exchange emails being redirected to GMail as well). I almost all other respects I love my Droid, and I, too, cannot switch to just GMail. It's too bad the native application won't handle these issues well, but K-9 Email is a better application! Good Luck.