Yes, they hear the conversation between VAR and the referee. They don't hear what VAR are saying between themselves, just as the referee doesn't, but once the button is pressed to communicate with the referee they hear that conversation. That's why they usually say there is going to be a review before the referee makes the signal, they hear VAR telling the referee they are going to recommend a review and are just checking the APP. This is the same in UEFA games as well.
As for Webb, firstly we don't know for sure he was listening to the audio, and if he was it was probably the same feed the broadcasters get. There's no way he was in communication with the VAR room or officials, and any suggestion he was is nothing more than fan led conspiracy theory.