In the Demo Mode (image attached), there's a transparent background behind the identity. Is there a way for me to set that in my own custom UI?
It would also be great if I could set a similar transparent background for the content in the center of the screen while on a call.
If you get into the phone web interface and into the Preferences->Appearance menu, you can see a lot of parameters under the Interface Colors settings.
In here you can modify some of the colours in the phone interface, keeping in mind that they are all defined by 4 parameters: the first one is the red value, followed by green, blue and optional alpha (that's basically the color transparency).
You can try modifying some of those, in your case to set the background color behind the identity as transparent, you need to modify this parameter "Selected Line Background Color" and you can set it to "255 255 255 0".
Unfortunately there is no way to modify the background during an active call but the background while on a call should already be transparent, we are working to develop a way to change it optionally but that improvement is still under development.