"¿Cuáles son nuestras fechas de nacimiento?"
Translation:What are our dates of birth?
There's a lot of confusion about "Qué" vs "Cuál", because people get taught incorrectlty that qué=what and cuál=which. Here is a better description... http://spanish.about.com/cs/grammar/a/que_vs_cual.htm
This particular example falls under the rule "Cuál usually is used before es and other forms of ser when not seeking a definition".
This falls under "weird things no one says". Unless you have a twin and you forget when you were born I guess. Anyway, I was hoping to get a little further clarification on "Cuáles" vs. "Que" because I think I know what you mean by open vs. closed, but am not 100%. Maybe an example of each would be helpful to be certain.
"fecha de nacimiento" is always "date of birth" (the thing you fill in on forms, etc), whereas "birthday" is "cumpleaños".
When referring to a "date of birth" you would always use "what" (not "when") because it is a piece of data, just like name, weight, height, eye-color, etc.
I'm always torn about whether translations should be equivalent or literal. You're right, in that the real question should be "When are our birth dates?" not "What," but then would you say that in spanish they're asking "when" or that they're actually asking "what" or even "which" but we would use a different word in english (when). Then if you're doing an equivalent translation, where do you draw the line for "interpreting"? Would you translate it as "When are our birthdays?" because that's what we would say in english (putting aside the absurdity of the question), or is that going to far, since the word you'd translate as "birthday" is actually "cumpleaño"??