Which Coveo for Sitecore version? Are you using the Coveo for Sitecore Hive framework or the legacy (WebForms or MVC-tagged) components? Is lazy loading enabled?

Which Coveo for Sitecore version? 3 or 4? Which CES version? Any CM/CD setup? If you have multiple instances, all your instances should have that field, since when a rebuild on your other instances will override this field. Also validate that the field is not duplicated in your ShowConfig.aspx (and the patch:before is unnecessary, the order in the fieldMap is not relevant)

This looks like another error, and a comment thread is not the best way to answer your question, you should probably enter a new question.

Have you tried updating the security cache for a specific role? In the Coveo Control Panel, under Actions > Update Security Cache > Update User Or Role. It will warn you that the user does not exist in Sitecore, but accept and it should remove the user from the security cache. Although it is still weird that this is happening in the first place... And the last error looks like a SitecoreMembershipProvider conflict error, does this instance have a custom provider that have colliding users?