Is a Muslim required to begin the obligatory prayer immediately upon hearing the adhan, and is that disliked? And is it permissible to rely on prayer timetables for those who do not hear the adhan regularly?

There is no objection to starting the prayer when the muezzin begins the call to prayer, because he only calls to prayer after the time for it has begun. However, it is preferable to respond by repeating the adhan. There is no impediment to relying on the prayer times recorded by trustworthy authorities.