Choosing a WiFi Range Extender with a LAN Port

It provides a wired connection for a nearby device such as a PC or console. Depending on the model, it may also support an access-point mode when connected to a router by Ethernet.

No. In range-extender mode, the LAN port can serve a client while the unit's link to the router remains wireless. Check the model manual for backhaul and access-point support.

It avoids a 100 Mbps port ceiling for wired clients, although actual throughput also depends on the extender's wireless link, router, client and internet connection.

Yes, if the extender has a compatible Ethernet port. The cable connects the PC to the extender, but does not remove the extender's wireless hop to the router.