WiFi Range Extenders with an Ethernet Port

It can connect a wired client to the extended network. On models with access-point mode, it can instead connect the extender to the router by Ethernet so the unit broadcasts WiFi from that wired link.

Yes, if the model has a port and supports the mode you need. Check whether it is intended for a client device, wired backhaul/access-point mode, or both.

A Gigabit port has a higher wired link rate than a 10/100 Fast Ethernet port. Actual internet throughput still depends on the extender’s wireless link, router, broadband and connected device.

Yes. Many extenders connect to the router wirelessly and rebroadcast coverage. Ethernet is useful when you want a wired client connection or a wired access point, but it is not required for basic wireless repeating.