I'm in general agreement with the above comments. Stay 4-5 nights in Reno, do the North Shore skiing from there, then 3-4 nights South Shore. North Shore skiing has more variety (and challenge, if Squaw is fully covered). South Shore is better for snow preservation, but that's more a spring issue, not as much difference in January.
Reno to Kirkwood on 395 and 88 is well over an hour but on usually uncrowded roads. I would normally do that from South Shore, but that might work on your transition day from Reno to South Shore.
I would ski Mt. Rose New Year's Day, closest to Reno and avoid end-of-holiday Bay area skier traffic, then Sugar Bowl/Squaw/Alpine Jan. 2-4 (no particular order, base on weather, conditions, mood of group), South Shore areas Jan. 5-7, keeping in mind the 7th is Saturday, might want to avoid Heavenly that day.
June Mt. is good for your move day Jan. 8 back to Mammoth, especially since it's a Sunday.