As to advice for shooting spherical panoramas, the most important thing is to keep the lens at the same point in space, rather than having it orbit around you. The closer your subjects are, the more important this becomes. Professional panorama photographers will buy special tripod mounts, such as those that GigaPan manufactures, which pivot the camera around the lens so that everything lines up. From that standpoint shooting order does not matter.
Ideally a panorama's subject would stay still while being shot, but this is rarely the case. The thing to remember is that you want the least movement possible, so if you have a human subject who moves horizontally it is better to shoot in columns, to capture them as near as possible at one location. If your panorama covered an exterior elevator which was in vertical motion, then shooting in rows is more appropriate.