Can I cool off the manicotti noodles completely before stuffing them?
What is your method for cooling the pasta? Is the goal to have it cool enough to work with? Or to stop the cooking process, keeping in mind it's going in the oven later?
My opinion is that the only cases when you should rinse pasta after cooking is if you're making a cold pasta salad, or if you're going to serve it plain. The reason why you should avoid this method of cooling it in virtually every other case is because in a great majority of the time you'll be adding a sauce to your pasta and you want the starch that's covering the surface of the pasta to help the sauce stick. This way all of the ingredients in your dish will be well incorporated and just give it that little nudge that makes all the difference.
Ultimately if you end up with overcooked pasta out of the oven you can always reduce the cooking time in the pot and if your sauce has enough moisture the pasta will keep cooking through in the oven (which would most likely be the cause of your problem in the first place, but we're clever, we'll work around it, take advantage of that!!).
I generally let shells cool 'til thy're comfortable to work with (running them uner cold water, then draining). I've never chilled them in the fridge, though. I don't know what that would go to their elasticity.
I would fear that for manicotti, they'd firm up too much and keep their crushed shape if they're not pliable enough when filling (thus, not acepting the same amount of filling).
If you're looking for a way to speed up your manicotti production, and hoping to split up the cooking the noodles and filling them to dramatically different times -- I've never had issues filling things the day before I bake them.
Another option for faster filling is to use crepes instead of noodles; you can still use a bag to pipe out the filling before rolling so that they're all of a similar size, but it's not as finiky as you don't have to shove it all in from the ends ... although making a lot of crepes can be time consuming.
Sure. Lots of recipes actually recommend that you let them cool:
Disclaimer, this is just a list of recipes. I have never tried them and do not necessarily recommend them.
Beef and Cheese Manicotti, Giada De Laurentiis:
Using a slotted spoon, transfer the manicotti from the pot to the oiled baking sheet and cool.
Easy Baked Manicotti, Kraft Foods
cooked manicotti shells, rinsed in cold water
(generally, rinsing pasta is not recommended)