In big airliners, does the flight crew hear the PAs the cabin crew is doing to the passengers, e.g. on their headsets? If not, how do they make sure they’re not talking simultaneously to the passengers?
Yes, when the pilot makes the selection to speak on the PA, he will hear if anyone else is using the PA. They would just wait until the other party is finished, or they can take priority and speak over them.
Some commercial aircraft can be equipped with an option to indicate to the flight crew that Public Address (PA) is currently in use (at least this option exists in the Airbus A330/A340 families, I am not sure for other series). In the A330/A340, the ECAM Memo section of the Engine and Warning Display will show a message "PA IN USE" in green color during cabin announcements.
In addition to the other answer by Mike Sowsun, pilots can not only listen to cabin PA announcements when they want to do one themselves, but basically anytime they want to monitor PA, by selecting the PA reception knob on their Audio Control Panel (ACP).
Most of the time though, this will be deselected because it is simply not necessary for the pilots. However, on the A330, it is usually intended that the third ACP (the one that someone in the observer seat would use) has the PA reception selected, such that the Cockpit Voice Recorder can record any cabin announcements on the PA.
All of this information as well as the image can be found in online resources like the A330 Communications and A330 Indicating Recording Systems sections accessible on Smartcockpit.