The trail will not be busy at all. There will probably be only a handful of people on the mountain if any at all. Your chances of getting a permit is 100%. There is only a quota from May 1 until November 1. Even with no quota, you won't have any issues this time of the year.
You will need full winter mountaineering gear. Not sure how much the axe would help as snow will probably not be consolidated. I don't want to give advice though because I have no clue what the weather condition would be in a month. I can tell you that the last time I went up in November, I was waist deep in snow at times. Gaiters, poles, crampons, etc would all be necessary as well as icy weather clothing...
Not sure what your experience level is but don't underestimate this one. Winds could be be +100 mph, subzero temps, etc. My November trip took +20 hours, basically doubling what it would usually take, and that was with a clear weather window.
