No - this product cannot be used with any lightstrips from a different brand.
Hi - you are correct - those images are more generic to the Philips Wi-Fi line of products. We will check to see if we can update those pictures that may have caused some confusion. Ultimately you are correct - the lightstrip does not screw in to a light bulb socket. It plugs in via a 120v AC plug and has adhesive tape that allows it to be mounted wherever you would like.
You can use up to 8 of the one meter extensions with the 2 meter base unit for a maximum length of 10 meters.
I think this should be okay, if you will cut the power cord, but keep the power adapter and the end plug. Then in between run a cable behind the wall. The only thing is to make sure that (1) the cable is rated for the voltage (2) The polarity after they splice is correct. If you need help, i would recommend contacting a licensed electrician for further assistance
The power cord is supplied for the full light strip. No power cord comes with the extension strip.