I want to make a lasagna quickly, and I don't want to be bothered with boiling the noodles OR going to the store. Would strips of freshly made pasta cook properly layered with the other ingredients? Could I use one large sheet of pasta per layer instead of several strips?
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've cooked with both home-made lasagne noodles, and with uncooked noodles (before they came out with the 'no boil' noodles ... you had to cook 'em for a good 90 minutes or so, and add extra liquid, as Marti mentioned)
Fresh pasta in lasagne comes out much differently than store bought noodles ... I grew up with it, but a few of my friends weren't thrilled with the texture.
And even then, we'd let it dry out some for an hour or two first, and then boil it, then assemble ... so it's not really a time savings over just using store-bought noodles (unless the trip to the store is such a problem). Also, rolling it can be time consuming, unless you have a pasta roller ... in which case, it's rarely the right size for the dish you're using, so you still need to use two strips or so to cut it down as you're working.
I'd say it'd be worth an experiment, but you're going to have to work quickly -- the pasta's not set up yet by cooking, so it liable to start getting soggy and dissolving if it sits too long.
Another alternative for faster lasagne would be to use wonton wrappers -- they're basically pasta in sheet form, undried. It might be faster than making your own, but you might need to double 'em up to get the right noodle to sauce ratio.