Do the size and shape of a cezve matter? 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:


The narrower neck of the cezve keeps the foam form. I haven't found any good references for an explanation; more discussion below.
It's easier to see what's happening. Using a bigger pot, it would be more difficult to see when the boiling/foaming is happening. Some stir it down during brewing, some don't; either way, timing is important!


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:


Moka pot. The size of the pot determines how much you brew at a time; can't really make more or less (though you could brew a few pots in a row). See also moka.
Espresso. Though you can make a single or double, and a long or short, you pretty much have to stop pulling at the right time.
Cone filter. For example, #2 and #4 and #6 coffee filters exist for a reason: it's hard to get a good pour of a 12-cup carafe of drip/filter coffee through a #2 filter; it's just too small.