Why does the Qlima™ portable gas heater shut itself off

If your carbon monoxide alarm is going off, you need to assume that there's actually carbon monoxide in dangerous levels in that area. If the heater is electric it probably shouldn't be emitting CO but maybe there's a manufacturing defect that's causing the heating element or plastic to burn.

CO is lethal and totally undetectable by people, so if the alarm goes off you need to leave the building and open the windows. The absence of smoke does not mean that you are safe.

(CO detectors have a limited lifespan: 2-10 years, depending on the model. There should be a date stamped on it somewhere. If it's at "retirement" age you should probably replace it anyway, and then you can see if the new one still goes off.)

Revision based on question update:

You must not ignore the CO detector. People can and do die in their own homes of CO poisoning, oblivious to the problem. Unlike smoke/fire, there's no way for you to tell the source of the problem or evaluate its severity, and no way to tell if the problem has been resolved. You probably have a sub-standard or defective heater — I would not use it anymore.

I don't think that you need to turn off the gas to the hot water heater. I believe that it is likely acceptable to leave the appliance on the 'pilot' setting.

I would obtain the manual for this device (try searching online, you will likely find it that way) or call the support number for the manufacturer.

However, what is acceptable practice is not necessarily best practice.

Personally, I would be hesitant to leave a gas appliance unattended for more than a day or two without physically disconnecting the gas (usually a 1/4 turn valve on the feed line). However, I feel comfortable lighting the pilot light myself.

Perhaps your should check with the local utility that provides your gas and see if they offer a service to visit your home and relight the pilot light upon your return? Mine does and it is free. My wife knows to contact them in the event I am unavailable because she is not comfortable at all in doing this herself.