Any advice on what the weather conditions might be backpacking the Mineral King area first week of October?

Cool. But don't take my word for it – I only hold one advanced degree in meteorology and have only been visiting MK for over 50 years.

KJR,
Mineral King is a terrific place,...if you've never been there, you will be amazed.
Early October in the High Sierra is quite unpredictable (like ROL said).
I wouldn't expect a foot of snow on the ground,...typically, that doesn't happen until November.
...Naturally, you should bring a good down bag, (plan for cold nights),...and, be prepared for snow flurries (and, possibly wind) at the worst.

KJR,
When it does snow in early October, generally it will be followed by blue skies within a day or two. Early-season snowfall adds to the overall excitement and beauty of the experience. Although generally speaking, accumulations will be minimal, there are exceptions. That said, I prefer snow to rain. Lightening occurs less frequently with snow as compared to rain, too. Bring an extra day's worth of food so you can hunker down while the storm passes in the event it happens near the end of your trip. It is easier and more enjoyable to hunker down than to attempt exiting the park in the midst of a storm. Stay dry and have fun!

I would worry about the road access if it snows heavily late September to your departure in October. The road to Mineral King could be the crux issue of the trip. Check with the RS in advance.