The power supply on the mini will be just fine. Supplying power to the connected devices will generate a bit more heat inside the case, but that's about it. If a fanless compact wall brick can deliver as much or more power to devices and work for years, there's no reason to doubt that the larger power supply in a vented case with a fan will be up to the task, especially when doing so only occasionally.
No, there is no up or down to the AC cord. If there were, it would be notched so that it only plugged in one way and one of the prongs would be wider than the other (assuming US outlets). The only role of the cord is to provide wires from the outlet to the mini with appropriate connectors on each end.