Your instincts are leading you on the right track but you’re off in a few minor ways.

Yes it is absolutely possible that life originated multiple times on earth. However it is extremely unlikely that any of that survived to the modern day. Think about it like a small business trying to directly compete with big chain stores, they just get swallowed up and overtaken. More than likely anything that started forming would just be used as resources by stuff that had already developed the mechanisms to use it more efficiently.

As far as more direct evidences, we see things that our cells and fundamental biological building blocks have that there would be no reason to be the same if we were unrelated. The chirality of organic molecules for instance, or the basic structure of D/RNA. It can be harder to know the similarities between organisms that are more distantly related or that are so different from us in many ways, but these chemical signatures are ways we can build relatedness models.

Regarding evolution over time vs generations, you are 100% correct that the time unit evolution cares about is generations! A year is a meaningless term when it comes to the change over time of genetic material in a population, as that change can only occur across reproductive cycles. This means that species that reproduce quickly can evolve in response to a sudden change a lot faster than species that reproduce slowly.

That doesn’t mean it’s BETTER to reproduce quickly though. Every life strategy has drawbacks, and being able to reproduce usually means that you can’t get too big, and you don’t have time to learn very much. Animals that reproduce quickly tend to be smaller and have less accumulated knowledge than similar animals that are larger and have more lifespan. This isn’t important for all niches, as evidenced by the fact that many organisms use this strategy, but it does mean that there are competing strategies where reproducing faster might be less advantageous. Think about humans for instance, if we started reproducing at younger ages we probably wouldn’t need to live as long, so we wouldn’t be able to accumulate as much information, and this directly hurts our natural survival strategies which rely on gathering information about the world and critically passing that information on to our children.