Chucks, whether in mortise or tenon mode need to nose up tight to the piece you want to hold. A regular woodworker will recognize the advantage of shouldering to resist racking.
As to sizing, if you choose to make a tenon, you will want to make it about a quarter inch over the first point of circularity on your jaws on green wood. The wood will shrink, and if you want to use the same jaws, you'll need some extra to re-turn to circular. If you choose to make a mortise, you may make it optimum diameter on green wood immediately, because you will be able to remove wood and make it optimum size again after drying. How large a set of jaws you choose depends on you. If you want a broad base, start there. If you want narrow, start there.
I like mortises, because they allow greater depth on the traditional half-log form than tenons fitted to the same jaws. I like dovetail jaws for either application because they don't "grip by corners" when in a non-optimum expansion, they wedge the piece firmly to the chuck for a long way over optimum size. Not license to be sloppy, but certainly not as big a deal as some would like to make of it.
You mention getting the angle right. It doesn't have to be perfect. It really only needs a gap a bit smaller than the wedge to be effective.