With most wildlife photography, our goal is to ensure the eye and the key facial features are within the depth of field and sharp.
Consider the ambient light and the colour / brightness of the species you are photographing. In sunny conditions, I’ve photographed Arctic Terns at f/13 thanks to their bright white feathers and the greater depth of field has helped with small focussing errors with this very fast and erratic moving species.
In many cases though, it is difficult to close the aperture sufficiently to ensure the whole bird is within the depth of field. Remember, we’re using long telephoto lenses and we need all of the light we can get to maintain fast shutter speeds.
Generally, I don’t worry about feet, tails and wings drifting out of the depth of field as long as the head is sharp. In many cases, I think blurring of the wings, caused by shallow depth of field or shutter speeds can provide a more dynamic feel to a photograph giving a sense of action and movement.
There is no absolute recommended shutter speed as different species move at different speeds. As a rule of thumb, I would try to keep shutter speeds at around 1/500 sec for large walkng mammals such as leopards and lions. That said, at dusk I’ve photographed a lioness and cubs walking directly towards me at 1/75 sec. It’s far from ideal though, I was relying on luck and this just leads to inconsistent quality and results.
For birds in flight we really need to be above 1/1600th sec but again, there are always exceptions and experience will become a key factor in helping you make this judgement.
Remember, if your shutter speeds are too slow then your autofocus settings and technique are pretty much redundant.
Don’t be afraid to increase your ISO. Cameras and post-processing software are so much better at dealing with high ISO compared to a few years ago. We can deal with noise easily, but rescuing a blurred photograph is altogether a different challenge.