No, nerves "broadcast" when enough neurotransmitters are received to cause an action potential. The action potential is a change in the chemical balance of the neuron that causes an electrochemical signal to be transmitted down the length of the nerve, resulting in the release of more biochemicals at the other end.
The speed is variable and is dependent on length and peripheral (e.g Schwann) cells
Yes to both in some cases. I can't recall specific examples right now without looking them up, but I'll be happy to GTFY if you need me to.
Which nervous system? Ours? Yes, I seem to recall reading that it is very similar to that of other primates (with the exception of the brain, of course).