The origins of when the earliest Immortals first learned of the Game, the Gathering and the Prize was never detailed during the course of the series, nor the books and comics. That was going to be the subject of "The Source", but the whole thing fell apart around 2005. What was known was that it was something that was undeniable and unavoidable. Ramirez had an inkling of what the results could be if someone like the Kurgan was the last, which is part of the reason he sought out Connor. And in turn, it is why Immortals trained other Immortals, rather than just always taking heads of the newbies due to the lack of experience and knowledge of their true potential like the Kurgan attempted with Connor. The connection to lifeforms was only present in the first film, as the sequels and series down played it. That's a common belief among the fan base as well. I tend to think though that it's more along the lines that if the Immortal is really evil, they will retain their Immortality and use their telepahtic and empathic abilities to their advantage in ruling. Good Immortals would become mortal, but use their new gifts to benefit mankind. Much like we see Connor do in the second film.