When I relist an item using RelistItem, and I've had either FeaturedPlus or Featured First as a listing enhancement, does my relisted item also have Featured First?

The short answer is yes. Relisted items pick up the original upgraded features.

Using DeletedFields with ListingEnhancements in the RelistItem request as 'Featured' will work to remove FeaturedPlus.

To remove the Featured First, seller can mark the Gallery Type as 'None'.
<PictureDetails>
<GalleryType><![CDATA[None]]></GalleryType>
</PictureDetails>

For a bundled site such as France, the item would automatically pick up Gallery.