1. “Cab driver, pull over at the next pay phone you see; I need to make a phone call.”
2. “Stay right by your phone and wait for my call, don’t leave even for a second!”
if I don’t respond to an after-hours call or a message until I’m back on duty, I cannot claim it as time worked.
That should go without saying.
That depends on the job itself, and/or whether or not such stipulations are spelled out when hired.
Any such stipulations can't override the law. So if the stipulation is that the person gets paid $x for those off-duty calls, that is fine as long as it is at least their hourly rate. EG: You don't have to pay them a minimum hour, per call. You can pay them at their regular rate for the actual time of the call.
But if the expectation is the person doesn't get paid, that would be illegal. (Of course the expectations for an exempt employee such as a manager or supervisor or different, and I'm talking hourly employees.)