Quenya wasn't the first elvish language. That is named Quendian. The first severing of languages came when some Elves decided to follow Orome (the Vala) from Cuivienen, the lake by which they first woke up. Those that decided to journey to Aman are called the Eldar, or wise elves, and their language was Eldarin, which is severed from the language of the Avari, who remained and modified Quendian further in their own ways.

Eldarin itself had subdivisions, and the much more populous Teleri spoke Telerin, which is the dialect of Eldarin from which Sindarin is derived. And remember, the Teleri were both the largest group and the slowest in their traveling westwards, so the division from the Vanyar makes sense.

Qhenya is a descendant of Eldarin, and I imagine going to Valinor and learning from the Valar had a great deal to do with adding words, and modifying existing ones that made the difference between Sindarin and Quenya sharper.

As for the Elves not dying... while that is true, they also didn't know much, in the beginning. There's a lot of the world they just simply hadn't seen, and thus had no words to describe. And since the various sub-branches of the elvish people discovered these things separately, and viewed them differently, it makes sense that the words they used for them was distinct, too.

All this is entirely distinct from how human language first evolved. There was no one time when a group of homo sapiens was suddenly born with the ability to speak. Language evolved slowly, instead, and there's no one instance where you can say there was a jump from sounds to words, and certainly no such transition within one lifetime. But within the mythology of Tolkien, it makes sense for language to evolve the way he lays it out.

As for men, the Avari languages they encountered, while different from Quenya or Sindarin, almost certainly had quite a few words and constructions in common with them. Enough that Finrod, who was quite the scholar, could pick up the language of these men quite well.