I can't re-order folders by just dragging them around. When I drop them where I want them, they default as a subfolder of the folder above where I want it to go (I already chose "Invent Sorting Order" within the folder where I wanted to shift things around. The only way to do it, which is a drag, is to drag each file up to the parent folder in the order I want it to go so that the first one I drop is on the bottom and the last is at the top. Am I doing something wrong?

Please pay attention to X coordinate of the mouse, it determines what will happen with dragged item. It will be a child if you move it to the right.

I have no idea what "x coordinate" is when it comes to a mouse, but I've tried everything and it's not working.

In other words, if to move mouse pointer to the right-hand side, it will increase item depth, and vice-versa. You simply need to play with it a little bit.