Web cams are showing snow sticking at 10,000'. So this hike just changed to a different animal. If you go over the weekend, you will be tramping through several inches of fresh snow. There is more of this storm coming -- it looks like a "tropical river" of sorts currently on the satellite images. So those several inches could add up to quite a few more.
You can certainly go, but you should take/wear gaiters (and extra socks), and if the snow has any depth, it will require a lot more energy. Be prepared to turn back if/when it gets tough enough that you can recognize that you couldn't make it in a day.
With the snow and cold, and the experience level of your gf, I think you have missed your opportunity to do the quick dayhike by one week this year.
As for White Mountain, I would assume, with this storm coming in, that Inyo N.F. has closed and locked the gate just before the Schumann Grove visitor center. They don't want vehicles getting stranded in the snow up there for the entire winter. Saw one up there like that a few years ago. You can call the White Mountain Ranger station in Bishop and ask about the road.
