Most probably where you are the crabgrass you have is the annual and it is done growing and producing seeds now, so your soil has a lot of them for next year. If at all possible find a source of a lot of compost to spread around, while grass would grow in a Lasaga bed the settling would be upsetting to the grass and you.
Fall is the best time to seed, but even in your area that is quickly coming to the end of the time to do it, unless this weather pattern holds. This link to a publication from Ohio State University USDA Cooperative Extension Service may be of some help.

Here is a link that might be useful: Natural Lawn Care

I don't know that lasagna composting will make a satisfactory lawn surface. A good lawn surface requires a lot of preparatory work to produce a smooth, stable, walkable surface. That means a lot of tamping and smoothing and more tamping. Lasagna composting is going to produce a soft but lumpy and squishy surface that compacts underfoot, and that has dips that will hold water instead of draining smoothly, and humps that will get scalped by the mower, and the whole thing will be annoyingly unlevel when walking on it and you sure as heck won't be able to play a decent game of croquet on it.

Now if what you really want is more of a meadow then ok. But the other thing to keep in mind is that that layer of compost on top of subsoil is not going to drain properly unless you till the two together.

I think my inclination would be to do the lasagna composting over the winter, and till and rake and tamp in the spring to produce the lawn seed bed, and then seed in the spring. You can try a cover crop on top of the lasagna over the winter I guess, I don't know how well those work in your area. I use crimson clover here but we have very mild winters, not much below freezing, with little if any snow.

The crabgrass seeds are already everywhere in your soil. They need pretty warm temps in spring to germinate so I think your best bet is to get the lawn grass seed going as early as possible to beat the crabgrass to the punch. Crabgrass is mainly a weed of thin scraggly lawns with lots of bare spaces so you can crowd it out.

For the hilly areas, depends on how steep they are. but you can get jute mesh that landscapers use for erosion control in just these situations. Do you really want lawn on a slope? maybe a nice shrub bed or woodland border would be better on your slope.

Well pnw7 that's a good point about the soft ground and I really didn't think of it mainly because I'm in a condo so I don't mow my own grass. Our soil is hard clay so it's real hard to blend anything in with it that's why at 1st the lasagna seemed like a real good idea. I was also told on the organic lawn forum that just throwing down compost on the lawn would help the grass to grow thicker so I guess I'll go that way instead.

Strouper