When brewing Turkish coffee in a cezve, it's generally recommended to fill with water up to the neck of the pot. Is it alright to use a bigger cezve (say, a double one) for making less coffee (e.g., a single demitasse)?
My guess is that the problem, if any, would relate to the fact that the water surface should be up in the cezve's narrow neck. A related question, therefore, is: does the narrowing-to-the-top shape of the cezve effect the brewing, and in particular, crema formation?
In short: Using a cezve of the correct size works better. The situation is similar to using appropriate equipment size in other coffee preparation methods (other examples at the bottom). That said, you can probably brew a passable cup of Turkish (or other names) coffee when your cezve isn't full to the neck (or, using a cezve that's "too large").
However, there are reasons for the shape and size of a cezve (as you suggest in your question), and it really does help in practice:
There are a bunch of brewing guides that suggest reasons; this one suggests stirring down several times, and says the size is due to "physics" without explanation; here's another. This one from CoffeeGeek suggests that the shape (narrowing toward the top) helps to prevent the foam from burning to the side of the pot (becoming bitter); this makes sense especially when using a small pot on a gas stove, when the heat will try to hug the sides of the pot. And this one doesn't say to stir. All of them suggest, but none that I could find explicitly state the reason why the foam forms better. In a pinch, I have tried to make Turkish coffee in a regular handled pan; it simply doesn't work as well. Something magical happens in the neck of a cezve to help that delicious foam to form; the best I can figure is that it's fragile, and the narrower neck somehow helps concentrate the bubbles, helps keep the foam together, and keeps it from collapsing.
Other examples of equipment size mattering when you're making coffee with other methods:
Proper size of cezve matters. Normally, it is ideal to fill up the cezve up to one third to the half of its height before beginning the brewing process. You need the rest of the height during the foaming time. You cannot fill it up to the neck and start brewing. In that case, at the end, when bubbles start to appear, it will spill over.
In Turkey, cezve is usually sold in sets. You can see a set of old style tin-plated copper set of five cezves in the image below. This set includes 1-cup to 5-cup cezves. However, I don't recommend them as the metal is not thick, so the heat is not homogeneously distributed while brewing. A second disadvantage is, tin plating could be worn away in time and copper oxidizes easily. The green colored copper-oxide is quite poisonous.
A modern stainless steel cezve set is also shown below. (Image is copied from Hisar cutlery firm's website.) This set includes 2-cup to 4-cup cezves.
As you can see, both sets include different sized cezves for different servings.
(A distinct advantage of possessing more than one cezve at home has another advantage: You may prepare Turkish coffee with or without sugar at the same time in two different cezves in case your guest wants to consume a different recipe than yours.)
So, I assume I make my point that the size matters and also the count is important. Let's talk about the shape.
Be practical, see inside. Check the spoon for stirring in the second image. You can see its length is ideal for stirring without burning your hand. Your hand will be over the vaporizing coffee most of the time. Find your ideal spoon, together with your cezve.
As it can clearly be seen, the shape is more or less the same. Wider bottom is mostly a result of finding proper balance on top of the stoves for small sized ones. You can see it when you look at them in the sets next to each other. If the bottom is wider, the neck seems narrower. When you have heavier stainless steel cezves, balance is already there, so the bottom is not that wider, thus the neck is not that narrower.
Still, I must admit that, a narrow neck is also related with the foam. The foam is formed by carbon dioxide packed in coffee lipids. When you heat the coffee, the carbon dioxide is extracted together with lipids and cannot be solved after some threshold in water. Those carbon dioxide changes its phase to gas form such as any effervescent, forming bubbles. This is the foam of Turkish coffee. If you try to brew Turkish coffee in a pan, the bubbles will be too far away and cannot survive in the surface of the coffee. They will soon pop and fade away. If they are close together, some irregularities of coffee (such as small particles in the coffee powder) stick on the surface of the bubbles and keep them together. Then, the bubbles form the foam. That's the reason a narrow neck helps to keep the foam.
