Are you an active member in the military? If you are, then you only have to reside in Wyoming for 90 days to qualify as a resident.
Otherwise it's a year ("12 months prior to applying for the license"). They will check TOO!
To answer your first question, hunting is superb in Wyoming. One of the main reasons I can't leave! Elk, Mule deer, White tail deer, Antelope (Pronghorn Sheep), Black bear, Moose, Mountain Goat, Rocky Mountain sheep, Mountain lion, migratory birds like goose & duck, pheasant, grouse, fishing for several species of native and non-native trout, walleye, cats, perch, bass, . . . my fingers are getting tired. All live and thrive here! And more public land than any other state in the lower 48.