Latin isn't "dead" in the way that many indigenous languages are dying, because as you note, it's still known, studied, used and taught. It's not even frozen: the Vatican has to create new vocabulary to express contemporary concepts and inventions, though that is an artificial exercise.

We don't know when Latin began to become Spanish or French or Italian, but it was probably pretty early. 4th, 3rd, maybe 2nd century CE? Tomb inscriptions are a good source of those changes. Those evolutions are both grammatical and word-based. Why did Italian (cavallo), Spanish (caballo), and French (cheval) take a rare Latin word for "horse, nag" (caballus) rather than the more common one, equus? No one knows.