As for wider versus narrower pots:
Yes, more heat is transferred into a wide-bottomed pot than a narrow bottom.
There are also more evaporative losses off a wider pot if you leave it uncovered.
Here are some dimensions of 1-liter pots (no freeboard at 1 liter so figure 800 ml usable) and their surface areas if lidless and lidded:
Diameter (cm), Height (cm), lidless surface area (cm^2), lidded area:
8, 20, 550, 600
10, 12.7, 479, 557
12, 8.8, 446, 559
14, 6.5, 440, 593
16, 5, 451, 652
18, 4, 477, 731
20, 3.2, 514, 828
If the wall thickness is constant*, then area is a proxy for weight. A lidless pot has minimum surface area/weight when the height to width is about 0.45, although anywhere in the range of 1.3 to 0.22 is pretty close.
A lidded pot, if the lid is the same thickness has minimum surface area with height:diameter closest to 1 and in the range of 2.5 to .45. Since most of us use lighter-than-factory lids, I’d suggest focusing on the lidless areas/weights.
*But wall thickness isn’t a constant. As the pot diameter increases, the pot gets less stiff and easier to dent, unless you increase the wall thickness. Consider a soda can and an aluminum pie tin. They have similar empty weights and volumes, but the can (small diameter, fixed lid) is much stiffer than a pie tin (very large diameter, no lid).
Stove efficiency would push us towards larger-diameter pots (and some have successfully used pie tins). Minimum pot weight would push us towards narrower pots (I’ve put heat fins on tall, energy-drink aluminum cans).
And, sometimes, like for Goldilocks, the diameter is “just right”. If your pot nests neatly around your water bottle or your butane canister, it takes up almost no volume and is better protected in your pack.