This recent thread may have some helpful advice: Let's Talk Lasagna With Fresh Pasta...
I am firmly in the pre-cook camp when it comes to fresh pasta in lasagna. The final texture is better and it is much easier to assess how wet the rest of your ingredients should be. That said, here are my answers to the rest of your questions:
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I would use whole-milk mozzarella, shredded from a block. Probably 1.5 lbs for a 9x13 Pyrex. I might sub in 8oz of mild provolone if I wanted a little extra flavor.
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I usually add an egg to the ricotta mixture, along with a good amount of grated parmesan cheese. However, I only use Calabro ricotta or my own homemade, which tends to be a bit thicker/drier than grocery store brands. Not a fan of cottage cheese in lasagna.
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Zucchini (and/or eggplant), spinach and mushrooms make a nice combo. I also like leeks, butternut squash and kale/chard. The key is to cook and drain them thoroughly before adding them to the lasagna. Squeeze all the water out of greens, roast everything else. Onions I would dice and cook with a bit of garlic rather than leaving in larger pieces. Carrots are an absolute no-go for me.
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In my experience, a 9x13 Pyrex can’t hold much more than three layers of everything.
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Pre-cook your fresh pasta.
