Why is a weight and balance graph lozenge-shaped? I have tried to find the answer but it seems to be really hard to find. Why is it not a square?
In a strict mathematical sense the graph should be extended into a complete right triangle, not a rectangle. However there are practical limits on how far forward you can fill the compartments so the graph can be cut off at 83 inches in this case. The top is clipped because increasing weight above that point reduces performance and safety margins below design intent and angers bureaucrats.
The reason for a right triangle is because the acceptable location of the center of mass is more critical at increased gross weight. When the plane is very low weight the center of mass can be very far forward and not overwhelm the tail down force and leverage. At high gross weights the tail needs more leverage to counter the nose down moment, so the forward limit of the center of mass must be brought back closer to the center of lift. There is no angle on the aft limit side of the chart because moving the center of mass behind the center of lift, at any loading, will make stall and spin recovery impractical or impossible on a traditional wing and tail layout.
The slight radial flair in the particular chart pictured is simply styling.
"The slight radial flair in the particular chart pictured is simply styling" is not quite accurate. This type of graph is the OEM's choice; many are in standard Cartesian coordinates. The reason for this style is for the purposes of loading in various manners. Regardless of the Gross takeoff weight, the fuel burn will be the same. If you create one fuel burn curve, this type of chart allows you to slide that curve to a new gross weight, and you will immediately see if the CG will go out of the allowable area. On a Cartesian chart, you have to plot every point and connect the dots instead of just sliding the whole curve. This is very useful on swept wing aircraft that have multiple tanks that are expended in a specific order. 707 has 7 tanks; a good example.
There has been discussion of moment and downforce capability of the horizontal stabilizer. Most of this is good, but you need to take that one step further. There are also structural limits at certain points of the airframe that cannot be exceeded. Too much downforce from the HorStab could exceed those limits. Conversely, a CG too far forward could exceed bending moment allowances forward of the wing root.
To build a chart, check out Vector charts in T.O. 01-1B-50 (2005), available online.