How can I sync email automatically without removing local messages that have been deleted on the server? Outlook on my computer deletes the server's copy of the message when it syncs, which unfortunately causes the Email app on the Galaxy Nexus to delete its local copy of the message when it syncs.

Is there a way to automatically download new email and receive notifications when it arrives without having it sync the deletions? In other words, once the email is downloaded to the device I want it to stay on the device unless I delete it from the device, but I don't want to turn off sync completely.

A related issue occurs when I've synced all the email in Outlook and the Email app on the phone deletes all the messages -- the Email app crashes when the Combined Inbox view is empty.  Is there any way to fix this?

(Aside: Is there a way to intitiate a send/receive of all email accounts from the Combined view in the stock email app?)
The behavior you described is what I'd expect from an account that uses both IMAP and POP protocols. Your emails are deleted from the server when you open Outlook and sync over POP but when you sync your phone over IMAP it keeps the mails on the server. 

You can get the desired behavior by switching your outlook sync protocol to IMAP. This keeps the emails on server and you have to manually delete them. Also, you should check out k-9 mail, it's a better replacement for the stock email app.