A sender app can send messages to change the state of the receiver app. Is there a cap to message throughput (other than network constraints)? Would it be feasible to send a live media stream using messages?
No, the data message channel is not built for that and is not going to do a good job if you try to use it for such purposes. The data channel is really a control channel, sending small messages infrequently to control the behavior, etc (it has a cap size of 64k if I recall correctly). Use other means (say a web server) if you need to stream media.