How formal is the Spanish word cuán? When is it appropriate to use, and when does it seem out of place? How are sentences using cuán normally expressed in informal speech?
In Spain, cuán is rarely used in normal, informal speech, though I think it would generally be understood.
To express the same, you can use qué, or lo ... que:
In questions, in Spain, it is often replaced by cómo de:
This also implies, in my opinion, that cómo de is also correct.
"Cuan" may be an unusual word for American-Spanish talkers but in Spain is quite common, at least in the "quantitative meaning" in questions. Other uses are not very very frequent and, usually, "cuan" is substituted by other structures (not by other words).
Cuan has two meanings that depend on the context, one of them is almost lost in the language.
1.- Cuan (in direct or indirect questions or in exclamations it should be written as "cuán"): Its meaning is "how much" or "how many" and it is a short form for "cuanto" (or "cuánto" in questions and exclamations). It is used when the word is just before an adjetive or and adverb ("cuan largo es", "cuan lejos vives") and never when you are making a comparison. It is very unusual to find it out of exclamations or interrogations but in old literature, and can be substituted by "todo lo".
Example: "Corro cuan rápido puedo"-->"Corro todo lo rápido que puedo"
In exclamations, it is usually substituted by "qué" so you will not find it except in literature, especially in poetry.
Example: ¡Cuán bello es el amor!-->¡Qué bello es el amor!
In questions, is more frequent, although is not the most frequently used form to ask the size of something. In Spain, it is very common to use "cómo de", and, in Latinoamérica, "qué tan".
Example: ¿Cuán largo es el vestido? --> ¿Cómo de largo es el vestido?//¿Qué tan largo es el vestido?
2.- Its second meaning is equivalent to "as" and introduces the second part of a comparison (as...as...). It is extremely rare, at least in Spain and in Mexico (I do not know other regions regionalisms), even in the literature. Almost always it is substituted by "como".
Example: Tan alto cuan feo--> Tan alto como feo.
I hope I have helped, although it has been quite a long time since you asked the question. I apologize for my English.
In questions, "cuán" is equivalent to "qué tan":
In exclamations, "cuán" is equivalent to "qué":
"Cómo de" is a regionalism. I think outside of Spain it sounds very bad. In general, if you can use "qué", this will be the preferred form in the informal and formal speech. "Cuán" is more common in writing, especially in literature. I think using "cuán" is not formal; it's just not usual. My hypothesis is educated people use "cuán" more frequently, but it isn't related to a formality issue.
Although this isn't really what you were asking about, maybe it's worth mentioning that it would be somewhat out of place to use cuán when you're in Mexico, where it is far more common to say qué tan. For example,