Have to add a comment here. Audio is always captured on film sets as Grumpy Mac demonstrates the process in his video. It is one of the reasons why filmmakers use a slate as it helps with syncing on-set audio with in camera audio or as a visual reference for syncing.
Many Hollywood films today are shot with multi-cameras, see Exodus Gods and Kings as an example. Ridely Scott said that using multiple cameras helped cut cost and speed of production. The reason why some films choose single camera only is because the director wants to monitor the scene. Since they can only monitor one camera at any given point they may choose single camera only. This raises the cost of filming as it means multiple takes of the same scene shooting for coverage from different angles and for different characters.
The question on audio monitoring is that you would audio record every scene you shoot. If you use a clip from one scene most of the time you would pick the audio from the clip angle you are using provided the audio is good. If it is not you could use the audio capture from another clip as the primary. If none of them audio is good from on set then you would go through the ADR process.
. A big key to shooting coverage is a consistency in blocking, placement of actors in the scene as well as the dialogue. if I am shooting a comedy where I know the actors are improvising lines, different on every take, I am more likely to shoot that with multiple cameras to get my coverage completely in each take. The less control I have of the performance or the more I want to give to the actors the more likely I am to shoot with more then a single camera.
There is no simple answer really to your question. Until you have worked with some multi-cam footage it is harder to explain completely. Once you get into the process you realize that what you do depends on the quality of audio captured, the method of shooting, and the way things happened while shooting. Software can assist greatly with the process by automatically syncing up audio with video between cameras.