Around what month do the majority of slopes and the peak usually open? Early January?

Depends on snowfall but most the mtn is fully ope. By the 1st

Although it varies widely year to year, over last 119 years of Big Sky weather history Nov-Jan have averaged 7 to 8 days of snow totaling 11 to 12 inches, risk of thin base snow depth in Nov, Dec is a safer bet, Jan should be fine in all but the worst years, view this site on desktop browser, click "All Data" at the top then scroll down to "PRECIPITATION - SNOW" section for details https://www.weatherbase.com/weather/weather.php3?s=440142&cityname=Big+Sky%2C+Montana%2C+United+States+of+America

I would avoid coming before christmas