1) rice is a great thickener, I use it all the time, but never in mushroom soup. Why? No good reason I've just never done it, should work fine if you are subbing it for roux.
I use mushroom as my thickener. A great stock I believe is what makes a great soup. I incorporate mushroom's in a lot of my stocks in general, think it adds dimension. Shiitake stems and or the dry kind being the most common. So if you have a good stick recipe just add some shrooms to the mix. I'd use vegetable or chicken stock, but veal makes a rich delicious soup just not very traditional, which brings me to my main question are you going for a traditional cream of mushroom soup or just a great mushroom soup? If traditional is the answer after the way I do it is I mix white mushroom caps (stems I would add to the stock) with one big Spanish onion, one leek, 2-3 shallots, and 3-4 cloves of garlic pr lb of mushroom. Mix it with enough oil to coat the mix. Put into a "hotel pan" (roasting pan) add some a 1/2 ts of Apple cider vinegar, TB of brandy, one TB of butter per lb of shrooms. Roast in a 375° oven for 1 hr (cover pan with aluminum foil)
When done add the contents of the hotel pan ( juice and all) to a soup pot add 1/2 a cup of heavy cream or lb of mushroom and blend adding your stock along the way to reach the consistency (thickness) you're looking for. Then taste and adjust the soup by adding salt, white pepper, and a little more butter for richness if you think it needs it.
For a "better" although not traditional soup I use a variety of exotic mushrooms to that mix and a half a cup of apple cider per pound of mushrooms note: still add the vinegar, I'm adding real apple cider also this time.
Happy Thanksgiving