I'm in Arkansas, and have been most of the last 60 years. As you said, if you are hunting in a Wildlife Management Area, you can carry a gun that is appropriate for the hunting season that is open. You are encouraged to shoot hogs in daytime if you see them; most of them are nocturnal now, so you may or may not have much in the daytime without dogs. Traditionally, if you use dogs, the only weapon you need is a knife.
There is lots of private land available to hunters in Arkansas. Timber companies generally have no objection to hunters, and there are still thousands of acres of timber company land throughout the state; most, however, is in the southern half of the state.
Be careful about the definition of a non-game animal. Several of the animals you named are furbearers, and there's a season for hunting and trapping them. To hunt them out of season, you will need not only the owner's permission (in writing, if he doesn't live on the land), but also a hunting license, and a special nuisance or predation permit from AGFC. Landowners may get by without the permit for predators and hogs, but it's risky. As a guest, I would bet you would need the permit.
On private land (with permission) you may hunt hogs day or night, 365 days a year. If I wanted to hunt with a big-bore air rifle year-round, I would cozy up to a nearby timber company, get written permission, and hunt hogs. If you haven't hunted them before, a few words of caution: they can run fast, they can hurt you, and they aren't particularly afraid of you. And they like corn soaked in diesel fuel.
