Can you add an element to a set without losing all previous data?

From the documentation it appears that the original Dublin Core Element set cannot be modified, only replaced. If you replace it with an identical Dublin Core Element set plus one new element, do you lose all previous metadata?

You can modify element sets; this is exactly what the Dublin Core Extended plugin does.

That being said, element sets are designed to be consistent from site to site, so the Omeka interface doesn't allow any methods for changing or adding element sets. Work with the element sets has to be done by a plugin (or manually, but that's not recommended).

If you're trying to add one element, and the Dublin Core Extended plugin isn't what you want, I suggest you think about using an Item Type to store your piece of metadata, or creating a new element set to hold it.