HDMI-CEC is a poorly implemented standard. Some devices, especially older ones, don’t play nice together. If it’s not working properly for you just leave it disabled.

Does your wife have another device at home that is logged into the same Apple ID?

The data on the phone is encrypted so unless the thief figures out the passcode no, they can’t get at the data, even her ‘dirty’ pictures. So, in other words, you are safe until you get home to deal with it properly.