Check the keylock in the driver's door. Make sure that hasn't been screwdrivered.
I've seen some vague reports here and there - nothing Cruze specific but it does seem to be possible. It doesn't seem to be widespread - yet. But I'd expect them to go stealing more valuable keyless ignition cars rather than fooling around with a Cruze. If it works like I think it does, you can minimize your exposure by not using the remote to lock the car. I find depressing the driver's door plunger as I get out works fine. Locking the driver's door locks all doors. But that same trick doesn't work so well on other doors.
Hmmm, I'm not sure how they'd change the frequency. I think there's only one frequency for any given county. What they could do is clear all the keys programmed to the car to make sure that there isn't a "spare" fob roaming around out there. Perhaps added by a malicious valet.
True. The Cruze is a computer on wheels. Much of the stuff is handled via a databus and remote modules. Even simple things like the door locks and dome light circuit. Killing power to a module is going to create more problems. (The BCM will throw a code if it's not able to talk to a module.) Frankly, the camera idea isn't a bad one. If you can get pictures of the guy, there's a good chance he's known to the police and they can go and find out what's going on. That's available in some dashcams and dashcams are not a bad idea.
I'll have to check what gets killed by the "battery saver" circuit. Because I'd be tempted to add a kill switch in series with that relay. I think there's a better chance that won't upset the system. (Although it's still a chance since the BCM wouldn't be commanding the shutdown.)
Entirely possible. The remote has quite a range. I can lock my car through two floors of cement and rebar. If the unlock button gets bumped, that could do it. I'd suggest taking steps to secure the keyfob from accidental key presses. There used to be a problem with uncommanded trunk release that was traced to the fob. GM came out with a change that required either two presses or a press and hold to release the trunk. But who's to say you couldn't have the same issue with the unlock? (Accidental button press) I'd also suggest not unlocking the car until you're right there at the door so you can verify if the car was locked before you arrived. If you're using the remote to unlock from 10 feet away, you have no clue if the car was already unlocked - so you have no "warning" that your car is unlocking itself.