Different kinds of Rabbits or Hare (Jackrabbit) have different habits. Jacks and Cottontail will dig a shallow bunker, sometimes in the open, mostly in the brush. European Rabbits dig burrows and do well in thorny thickets, so the Fox and Yotes have a hard time digging them out. Don't discount Jackrabbits, they are thin and built for speed, but are actually tasty.

If you have grass, Clover, deep brush,hedges or thorny thickets you will likely have Rabbits. Some Alfalfa plots will also help. They seem to thrive where deep thick cover meets abundant food.

Pheasant seem to prefer tall weeds or hedges, but have learned on my lease to nest in Blackberry (thorny) thickets.

Your woods sound like good Turkey habitat, but you'll have to ask somebody more knowledgeable than I.

Small game draws predators. Thinning out the predators can be man hour intensive. In an average year my (personal) small game harvest and my predator count is about equal. "If you build it they will come" and so will the predators.

Deer often prefer someplace elevated to sleep, often hedges or deep brush with a view.