Heather, you can hike beyond the "undocumented zone" in a day or two. If you choose your campsite away from the trail, are quiet and don't build a fire, there is relatively little risk of encountering illegals or drug smugglers. If you are aware of the possibility and keep your eyes open then there is not much risk; they try to avoid hikers.
Most of the Arizona Posse types don't get more than 100 yards from the beer coolers in their pickups and thus are not much of a risk.
In most years January here is sunny eight days out of ten. Most days are in the 50's to 70's and nice and sunny; nights are cold (not by Fairbanks standards though); it can get as cold as 10°F in higher locations at night and into the 20's in the desert lowlands.
There are occasional frontal winter storms that come into the area with rain or snow but these seldom last more than a few days and are not usually of the sort that keep you from hiking. I would not leave my rain gear behind.
May I walk in beauty.