Is it because that the distance 13m are too long for the two switches to communicate? Or because of the quality of the cable?

ah, that is pretty hard to say which store sells the V3 version. I think you can go to theTP-LINK website and seek for where to buy, check with the retailers who sell TP-LINK product wither they have...

TL-WPA281 can be both wireless and wired.

How about reset the router? Perhaps you can also try to use a static IP address on the client.:)

As to the second one, I know that the DHCP list will only show the device that get the IP address from the router by using DHCP, if they have a static IP, the router won't show them. In that case,...

If it only concerned with the wireless network connection, you may try to fix the channel on the modem. Or give a try to upgrade the firmware, reset the modem and reconfigure it.