What Smellytele said about the car. The alternative would be to rent an AWD/4WD vehicle, but over the course of three weeks, the chains will easily pay for themselves. If you're a Costco member, I'd check with their tire center (775-689-2235) to see whether they carry chains; they're literally across the highway from the airport. (Obviously, it's also a great place to stock up on prepared foods for your kitchenette.)
I assume that the Motel 6 reservation is not pre-paid, and you can cancel without penalty? Assuming that's correct, I'd also continue to check Reno, Sacramento, and Bay Area Craigslist, as well as VRBO, for lodging options closer to Kirkwood. Other than President's Week, you won't be staying during high demand time, and private party rentals should be willing to effectively throw the weeknights in at nominal cost.
If you're thinking about dividing your trip between the south (Kirkwood, Heavenly, and Sierra-at-Tahoe) and north (everywhere else) areas, I'd suggest dividing your hotel stay as well. There are good budget lodging options on the other side of the lake as well, and you don't want to have to drive around the lake on a powder day.