l. . . . Could also be looking for something that receives wi-fi and provides a local ethernet connection that could then be plugged into a normal wireless router
For this approach, an inexpensive and robust method would be a dd-wrt capable router used in client mode like THIS
with another inexpensive router attached to one of its LAN ports in AP mode to provide the wireless radio. I expect that this would be a lot cheaper than buying a device that will do both in one box, although that is an option.