1. Are there keyboard shortcuts for this? In Photoshop, when you select an element or a layer, you can use the arrow keys to move it. I remember being able to do this in really old versions of Premiere, but I don't remember how.
2. Is there a way to keep elements inline when rearranging? In Photoshop, you can hold the shift key and when you drag a layer, it will stay on an even axis (horizontal, vertical, 45-degree angle).
What "Program window" are you referring to? There is no specific "Program window" in Premiere Pro; there is the Project Panel and the Program Monitor. Are you talking about one of these?
Assuming you're referring to multiple video elements in the scene, which can be manipulated using the mouse directly in the Program Monitor, I'm not aware of any Photoshop-like functionality. The closest you might come is to adjust only the x or y axis in the Effects Controls Panel.