@BillL907 

The ethernet port on a Range Extender serves up to 2 purposes.  1. To provide a physical connection point for devices that do not have wireless to connect physically to the extender to connect to the internet.  2.  If the Range Extender has an AP mode it acts as the phyiscal connection point to the router and provides the wireless signal.  For your questions it should have no issues working as long as you are in Range Extender mode.  Your senario will not work in AP mode.  Also make sure the Range Extender is in an area with a good signal and between the router and the area service is needed.

Best Regards