I've heard many differing opinions on leaving leaves on the lawn. Everything from "leaving them will keep your yard green" to "leaving them can kill your grass."

Thus far I've been collecting the leaves and dumping them in the woods on half my property. I've also mowed over them to mulch them, but I heard that that can disturb overwintering insects.

As far as I'm aware, I basically have 4 options:
1) leave them where they fall
2) mulch them
3) compost them
4) dump them in the woods

The least labor intensive and most environmentally friendly option would probably be #1, but I have a small yard with a large tulip tree and hickory which drop lots of large leaves that do tend to smother the grass. I'm in the process of getting rid of the grass and replacing it with native plants, but I'm worried those will get smothered too!

I'm interested in everyone else's experience with this. What do you think is the best method?

2/3 or 4

If you want to kill the turf... then, yes... leave them where they fall.
The locally native perennials evolved to cope with dropped leaves...
If you are growing plains plants... then, maybe not leave them covered as they aren't woods plants.
Always best to be specific... It's fairly common to call everything that grows in the US "native" when it's not really correct to call something that evolved in different conditions "native".

I'd go for 3, or add a 5: pile whole leaves in garden beds, corners of the yard or directly under trees. The woods already have their own leaves. Why not put yours to use in your garden? Leaves on manicured lawns can cause problems with matting & smothering, but you're also getting rid of it, yes? Once you get plants in you can re-evaluate...that could depend on the growing habits of the plants.
As for the insects, that's a big consideration. Here are a couple of articles that might help you decide.

https://www.xerces.org/blog/wh...

https://www.xerces.org/blog/le...

We tend to mulch small amounts of leaves but I like to pile leaves up to a foot deep on my flower/ strawberry beds. This year I hope to run the mower over leaves and bag it, then use the chopped leaves on my flower/ strawberry beds and a decent layer on my garden since I don't have room for a compost pile.

I like to make a leave Compost Pile....It is worth it when the leaves break down & turn into nice leaf compost. There are no weed seeds in the leaves so that is a big plus when using them. They also do not take too long to break down. Turning them occasionally helps, but I know not everyone has a tractor to do a fast & thorough job of it. Thanks

Definitely if you leave them on the lawn, the grass will eventually have trouble seeing the light. (At least, such is the case with the maple leaves we get lots of every fall!) What i like to do is collect all the leaves I can use as overwintering mulch under and around my container plants (just the 2nd year doing that), and then mulch the rest into the grass as we've done for many years. The leaves clear right up the moment we mulch them in! And I'm sure it's much better for the lawn underneath. Too bad all these fallen leaves aren't so convenient to spread to the open treeless area of the yard to mulch into it there, though I think that's a good idea to do as well.