We don't know if Hezbollah was the only customer. If so, it was probably permissible in response to a proportional attack (like lobbing rockets at Israel). I would think, anyway. (IANAL) If Hezbollah was not the only customer, that could be a problem.

I used to have to carry a pager. Pretty sure it worked off the cell network. Not sure about these, but modern aircraft often provide cell service in-flight. If these devices could be pinged in-flight, they would have been a hazard. And if a customer was a hospital, that would really be a problem. Did Israel manipulate it such that Hezbollah was the only customer? So far I haven't seen anything one way or the other.

I believe, IIRC, that may old pager would queue messages. So if it was off, or I was someplace shielded, the message alerts would come in whenever a signal was restored. Assuming that is how they triggered it (i.e. a message from a specific number).