The drillstring is generally run box up/pin down because it is easier to stab the pin connection into a stationary box connection. Generally, your point of view is above the connection so stabbing the pin into the box allows the operator a better view of what is going on. This results in faster trip times and less damage to the connections.

If for some reason, you had the connection above your line of sight, I guess the standard practice would be pin up and box down, but that is just not how a normal rig operates.

As for the connection types used in the Bottom Hole Assembly (BHA), yes, you are correct in that it is to minimize fatigue. The connections in the BHA are nearly always run in compression (as opposed to those in the drill pipe string which are under tension). The portions of the drillstring that are under tension tend to rotate about their own axis (at least as much as they can) as the tension tends to pull the string into alignment. Under compression, however, buckling can occur which can force the connection away from the center of rotation. When the axis of the connection is no longer aligned with the axis of rotation, the alternation forces due to bending are increased. In addition, the connection OD of the tools in the BHA can change dramatically while, in general, the ID of these tools is the same. The stiffness, or bending strength, of the box connection is dependent on the OD and the thread characteristics (lead, taper, thread length, etc...). The stiffness of the pin is dependent of the ID and the thread characteristics. To minimize the fatigue damage, you want to balance the bending strengths of the box and pin. This balance is known as the Bending Strength Ratio (BSR). See API RP7G, or other drilling handbook, for how to calculate the BSR and what range of BSR you should consider for your particular application.