This could all be wrong/speculation but I'll take a crack at it from what I know.
1. I've read that mostly generals used the Voice, only to shout down the gates of a city during a siege.
2. I assume that they learned it how the Graybeards learn it, through practice and meditation. The old Nords were probably more attuned to it because it was used more than it is in the fourth era.
3. It fell out of use sometime in the first era. The tongues fought the Dwemer and Chimer on Red Mountain and lost. Jurgen Windcaller, one of those defeated tongues, pondered the loss. After 7 years, he realized that the gods of punished the Nords for misusing the Voice, and thus the Way of the Voice was born. The other 17 defeated tongues became the first Graybeards and they're pretty much the only ones that used the Voice from then until now.
4. It would be possible for all Nords to learn the Voice. The process would be much like how Ulfric learned it, practice and meditation and those such things. The effects would probably upset the Graybeards, and as for the balance between the races, it probably might make the Nords more powerful. If the Voices are the same as lore, then yes, the balance would be very upset haha.
5. Battle Cry is similar to the Voice in lore, so yeah it is kind of. The Voice and shouts and whatnot became muddled when Skyrim came out because instead of it just being similar to a battle cry, they became the language of the Dragons.