pres is trained to know that they must hit play on the news ramp in at 51 secs because news reader will begin regardless. They use the clock in the studio to know the time and all the songs will be scheduled to account for this - if they are a minute or two ahead of sched they'll fill.
as for show handover, lets say Greg is on Air in 82A and RMC are going to go from 82B - 82A has control of the network, shortly before RMC are due on air, they'll output a test tone. 82A will assign the output of 82B to a fader that they will then listen to, checking for the test tone.
If all is okay, 82B will stop the tone and 82A will fade them down. 82B will have the output of 82A on a fader of theirs so they can hear the studio and their cue to start about a minute before their on air time, 82A will fade 82B back up, and the first part of the show from 82B will go to air via 82A.
When everything is okay, 82B will request control of the network on the TX system, which 82A must approve as the currently on air studio 82A would the release their control, handing it over to 82B who is now the on air studio.
The "robot clara" effect you sometimes get occurs when a loop happens. If 82A is in control of the network, theyll have 82B faded up, but if 82B has 82A output faded up to their output, it loops back so you get this loops of audio that results in the robotic sounding audio.