Dear @PierreMatteau,

Wondering if it possible to connect (via Cat 6 cable) a EAP 245 to my Apple AirPort Extreme for the purpose of extending the WiFi network created by the AirPort  Extreme?

Yes, it is possible. The EAP245 is a wireless access point that turns wired LAN into wireless LAN.

Right now I have both units set up as individual WiFi Networks and I do not find this very elegant as you have to select which Wifi Network to connect to depending on where you are in the house...

Perhaps you can try setting the EAP245 with the same SSID and password as the existing WiFi.

As an additional point, if I need to provide power to the EAP can it be setup using PoE using PoE switch?

Would the connection sequence be: Internet into Airport Wan connection, Cath 6 cable from LAN port of Airport to PoE switch, PoE switch Cat6 cable into EAP 245?

Yes and yes.