"What time is it?"
Translation:¿Qué hora es?
·Cuál· is used before forms of "ser" when not seeking a specific definition: ¿Cuál es tu problema? (What is your problem?)
·Cuál· is also used to when suggesting a choice from a group, list or selection: ¿Cuáles quieres? (which ones do you want?) Also note, if you were to say ¿Qué quieres? it would be suggesting what does someone who is talking to you want from you, rather than a list of items.
It helps to think of ·Cuál· as "which one" when you use it for "what"
·Qué· is used when seeking a definition. ¿Qué hace un mestro? (what does a teacher do?)
·Qué· is most often used before nouns (however, using cual isn't always incorrect either. So it's kind of tricky at times) ¿Qué libro tienes? (What book do you have?)
And ¿Qué hora es? is actually an idiomatic expression, instead of saying something like ¿Cuál es hora?
This should help answer all your questions:
What is the distinction between hora and vez in this context? Both have "time" Listed as a translation.
My own guess based on the way the words have been used in problems is that vez means "time" in the sense of how often something happens and not in the sense of when it occurs.
Is my interpretation correct? Are there further subtleties of meaning that I've missed?
Adding "ya" is the same as saying "now" or "already" and has a wide variety of uses based on context. If you know the word "todavía" (which means still), the main difference is "ya" indicates completeness while "todavía" indicates something ongoing.
In your example, it would translate to "What time is it now?" and suggests emphasis that you really want to know the time right now.
An example I hear often with "ya" is 'ya veo' (I see). In this example "ya" suggests that you have a thought or wish satisfied. Maybe like saying "ahhhhh now I see".
Here is a more thorough explanation of when to use ya
The "temporary" aspect of estar is obviously too simplistic, because it doesn't work with time, permanent locations, nor in the quite permanent case of está muerto.
Instead, use the DOCTOR rule for ser: Descriptions, Occupations, Characteristics, Time, Origin, and Relationships.