Can a lot of the practical points you raise be answered by asking an experienced musician (perhaps, especially a mridangist), "If you hear a completely new-to-you piece of music, can you determine the exact tala, especially when the number of counts per cycle is not, in itself sufficient to to provide the answer?"
At the most extreme end of complexity, we know the stories of old: of the mridangist telling the palavi singer, "if you will not show me the talam, instead of keeping your hand hidden, I am going home," but how about the ordinarily-complex levels?
I have always thought that they seem to be able to do so --- but I might be wrong. I wonder what msakella's response to this is, not only from his own experience, but from a lifetime of knowing many, many musicians?
I guess all I have done is re-phrase your point (3)