How do I populate an email field from a single user “Name” reference?
Maybe I don't understand the need, but once you reference an entity what is the point of adding the values from the referenced entity into the referencing entity?

Is this a UI thing when creating a node? Or a display issue when displaying a node?

If it's a display issue, wouldn't you just create a new display for the reference user for when it is being referenced only showing the fields that you want shown?

To do what I am saying, you would need either https://www.drupal.org/project/ds Display Suite, or embed a view, or call the referenced user fields in a template or module.