well, carb is a lot bigger than a lawnmower's anyway. I would go back to the choke if it's not working, but try it first. On the t.stat -- should open around 140F, give or take. I heat up a cup of water in the microwave and test them when my wife isn't home. Carb rebuild? Need a kit, and recommend full disassembly and cleaning -- soaking everything metal, and going over every opening with some carb spray and very fine wire. Others may take a different approach. (Need to take off and replace the expansion fitting on the top, though, and clean out the passages under there.)
edit. I might not be understanding what you are describing with the key -- might not be an appropriate one though, whether damaged or not. I've reused them, but sheared a couple working on a similar motor, so take some special care there I guess.
edit2. When shopping for a carb kit, good to get one with a new float -- not mandatory, as the current one can be reused or sealed and reused (above), but the non-cork floats often recommended. [poss. you already have one, if PO did a rebuild]