Yes, you can connect a wire to transmit the wireless signal from a wireless router.
Yes, you can connect a wire to transmit the wireless signal from a wireless router.
According to others, the gadget shown in the picture lacks any wireless function.
The issue is that WiFi loses a lot of signal even with good cables. Most routers use multiple antennas, so you need at least two for best performance. Building that setup would cost more than simply running Ethernet into another room and adding a dedicated access point there. The result would be significantly improved speed and reliability.