Warren in Lancaster, PA
There is no problem using green wood for this. I recommend splitting a branch down the middle or into quarters to get nice stock. A piece of a branch that contains the pith is vulnerable to heart checks. If the pith is near the edge of your stock, it will probably split from the pith to the nearest edge, but will otherwise be sound.
An apple branch that grew at an angle will have compression wood on the underside. This is a form of reaction wood. The tree makes extra wood on the underside of the branch in order to counteract the tendency for the branch to droop and there can be three or four times as much wood on one side of the pith as the other. Also such a branch can distort a bit when it is cut open.
Green wood cuts cleanly with sharp tools; I don't recommend filing or sanding.