Snow, in and of itself, doesn't really change anything with regards to the sleep system.
10*F is 10*F with or without snow on the ground.
You do need a tarp for the snow but I am assmuing you have one already.
You do need to keep the snow out of your boots [gaiters or high boots].
You do want to wipe snow off of your body/clothes/gear fairly frequently since it will melt due to your body heat.
Love hiking with a gentle snow fall. Magical.
High 20s low 30s with wet, slushy, snow and icy condtions are tough. I usually stay home or just dayhike if those conditions are expected.