I'd go with the ideas you have already.
Simmer the carcass in water to barely cover. (If there is meat on the carcass that is removeable, do so; if the meat is cold and less easily removed now, simmer the carcass for several minute to warm the meat, remove the carcass, remove the meat when cool enough to handle, then return the carcass to the pot.) Cook till the carcass has given up its flavor, probably an hour or two, then remove it. Taste the broth and reduce as needed to concentrate the flavor.
Add some finely diced carrot and celery to the pot if you wish.
Meanwhile, is a large sauté pan, sweat onions till tender or, better imo, caramelize them lightly. Add a little garlic if you wish for the last minute or two. Add a little thyme. (A TINY bit of ginger is a good addition; not enough to be noticeable as ginger per se, but it adds a certain something.) (If wanting to add, say, finely diced red bell pepper, green pepper or hot pepper(s), add to the onions when they are just starting to brown in spots.) Dump into the pot.
Bring to a simmer, add pepper to taste, a little salt, if needed, and perhaps a small bay leaf.
Sauté your sausage at whatever point is convenient so that it browns nicely on its cut sides (better flavor). (A little shredded chard would be a good addition is going with sausage and wild rice.)
Add the wild rice to the pot and cook till it is almost tender. Add the cooked sausage and whatever turkey meat is available. Cook just till the rice is done. Serve, drizzled with a little malt vinegar, aged sherry vinegar, slightly reduced balsamic, or whatever vin you fancy.
