1. Is there an ideal ratio (length to width to height) to maintain in designing a box so that it is pleasing to the eye?First I will say that there are not any definitive answers to either of these questions. There are so many considerations in making boxes that there are no absolute right or wrong answers. On question 1. some box makers and furniture makers are intrigued with the golden ratio, a system of proportion based in the fibonacci sequence of numbers. Personally, however, I have made most of my boxes with an eye toward their use. It may be that some shapes and sizes will have particular appeal, but I would consider it an unreasonable constraint to impose a system of proportion on my making of boxes. Even if there were an ideal shape, you would have to stand in just the right position to see it.
2. The stock that I buy from the wholesaler is usually about 15/16 inches thick. Does it make sense to try and get two equal thickness boards which probably will wind up to be 3/8 inch thick or should I just bite the bullet and make one piece over 1/2 inch so that when I plane it down it would result in a board about 7/16 inch thick and the rest is basically put into the scrap box and used for small dividers and trays?
On question 2. I find that smaller boxes feel best if they are made with resawn stock. Larger boxes feel best if made with thicker stock. This will mean that for small boxes you will want to resaw your stock down the middle and use both parts for box sides. If done well, you can get 3/8 inch stock, though I use 5/16 inch material for most of my small boxes. For medium size boxes, you may do as you suggest and use offcut material for dividers and the like. For large boxes, you may just want to plane the material down to the desired thickness. There are also other considerations. For instance, what kind of hinge are you planning to use? Hinges, because of their design may require that box sides be a specific thickness to feel right.
As my readers can see from the photo above, Nestor is doing quite well in his box making. I like the inlay he applied to the top panel.