Won’t fresh pasta sheets in lasagna overcook

Like Bob said, surely boiling some noodles is way less of a bother than breaking out the pasta machine?

If you really want to make fresh pasta, I don't see any reason you couldn't use it for lasagna without boiling it - the reason one boils dry noodles is because it's hard to get them to soften in the time it takes to bake the lasagna. But just like when cooking fresh noodles, you need to be careful not to overcook it, so make sure all your fillings are fully cooked before you start layering. It also should be perfectly fine to not cut the noodles into strips - the reason dried noodles come that way is for easier handling.

It is possible to make lasagna using dried lasagna noodles without boiling them first, but you need to add more water and bake it longer. A search for "no-boil lasagna" will turn up scores of recipes.

I have the same preference. Three things that work for me:

(1) If you are using any vegetables, saute them to get as much liquid out as possible in advance. For spinach or other greens, actually squeeze the water out using a tea towel or potato ricer.

(2) Use less sauce or make the sauce thicker. You can accomplish the latter by starting with thicker ingredients (tomato puree vs. chopped tomatoes), or by reducing it on the stovetop somewhat first.

(3) For ricotta fillings, indeed add a couple of eggs to help them setup.

Here is my spinach and ricotta lasagna recipe, which as you can see from the picture, sets up very firm and sliceable. And I've also got a summer squash and portabello mushroom lasagna which illustrates points 1 and 2.