Meat – Do I need to cook the meat filling before stuffing ravioli

It's neither necessary nor a particularly good idea; it does little to remove bacteria from the surface of the meat (which you're about to cook, remember) and runs the risk of spraying/dripping bacteria all over the kitchen.

The FSIS has an article on it here:

Washing raw poultry, beef, pork, lamb, or veal before cooking it is not recommended. Bacteria in raw meat and poultry juices can be spread to other foods, utensils, and surfaces.

The only exception would be something like vac-packed, wet-aged meat where you need to remove the salt, but that's a matter of preference.

I have never washed a piece of meat in my life and I'm still here!

You may already know that pastry making requires everything to be cold - the fat, the water, your hands, the board, everything. This is to prevent the fat from melting into the flour prematurely, which results in cardboard-like pastry. When the fat remains cool until cooking, it melts into layers in the pastry as it cooks, making it nice and flaky.

If you put hot filling into your pastry case, you will melt some of the fat prematurely and so the pastry won't be as good as would be if you let it cool.