Yes, cook the meat before stuffing into the ravioli. If you are worried about your mixture being too fatty (which I didn't experience with a non-traditional beef and bacon ravioli), make sure to drain the meat well after cooking, perhaps patting it with clean paper towels to remove excess grease.
Lasagne typically need to cook in a hot oven for about 30 minutes. The main problem, as outlined by others in this thread, is the tendency of pasta sheets to dry up during this prolonged exposure to high temperatures.
To avoid this, I usually precook the sheets in slightly-salted boiling water for one minute (I just want to soften them, not cook them); a few oil drops in the boiling water should help preventing the sheets to stick to each other. Subsequently I drain the sheets and lay them on a clean cotton cloth to let the cooking water be absorbed while I attend the preparation of the sauces.
After lasagne are assembled in the pan, I cover it with aluminium foil and then put it in the oven to cook; after 20 minutes I remove the foil and let lasagne cook "naked" for the remaining 10 minutes.
Here are some pictures I took during the preparation of lasagne with crumbled sausages and mushrooms.
With this procedure, lasagne sheets retain most of the moisture, thus not needing an excessive amount of sauce to keep them hydrated.
I managed to obtain very good results even with Sainsbury's Value Lasagne Sheets (a brand of cheap "no precooking required" dry lasagne sheets commonly found in UK).
Best Answer
Time...lasagna needs to allowed to rest for a while before serving. At least an hour. If you try to serve it straight out of the oven it will slide all over on you. Time will allow the cheeses and other filling to firm a bit to give you the distinct 'layers' that you want to see out of a traditional lasagna. I would even recommend making your lasagna the day before you plan to serve it and chilling it over night and reheating just before service.