Place dried porcini mushrooms in medium bowl; pour 1 1/2 cups boiling water over. Let stand until mushrooms are soft, about 45 minutes. Using slotted spoon, transfer mushrooms to cutting board and chop coarsely. Reserve soaking liquid.
Meanwhile, heat 1 tablespoon oil in heavy large skillet over medium-high heat. Add crimini mushrooms and garlic; sprinkle with coarse salt and pepper and sauté until beginning to soften, 2 to 3 minutes. Add 1/2 cup wine and simmer until crimini mushrooms are soft, about 4 minutes. Set aside (there may still be liquid in skillet).
Heat remaining 2 tablespoons oil in heavy large pot over medium-high heat. Sprinkle pork with coarse salt and pepper. Add pork to pot and sauté until browned in spots, about 6 minutes. Using slotted spoon, transfer pork to medium bowl. Pour off all but 1 tablespoon fat from pot. Reduce heat to medium. Add coppa and stir 1 minute. Add sausages and cook until brown, breaking up into small pieces with back of spoon, about 3 minutes. Add onion, carrot, and celery. Cover pot and cook vegetables until soft, stirring occasionally, about 8 minutes. Add remaining 2 cups wine; bring to boil, scraping up any browned bits. Simmer until almost all liquid is absorbed. Add tomatoes, 1 cup broth, bay leaves, reserved pork, and porcini mushrooms. Pour in reserved porcini soaking liquid, leaving any sediment behind in bowl. Bring to boil; reduce heat to medium-low and simmer uncovered until pork is tender, adding more broth by 1/4 cupfuls if dry, about 1 hour.
Stir crimini mushroom mixture in skillet into ragù. Season with salt and pepper. DO AHEAD Can be made 1 day ahead. Cool slightly. Chill uncovered until cool. Cover; keep chilled. Rewarm before continuing.
Spoon off fat from surface of ragù; stir in basil. Add Potato Gnocchi; toss gently to coat. Simmer over medium heat until gnocchi are heated through, 3 to 5 minutes.
Divide gnocchi and ragù among bowls. Sprinkle with some of cheese and serve, passing remaining cheese alongside.