"What day is your birthday party?"
Translation:¿Qué día es tu fiesta de cumpleaños?
The what/which distinction is only a rule of thumb.
When used before a noun, "qué" is more usual . "Cuál" should also be accepted, but it tends to imply a more limited choice, e.g:
"Mi familia llegará el jueves o el viernes."
- My family will arrive on Thursday or Friday.
"Cuál día es tu fiesta de cumpleaños?"
- Which day (of those two) is your birthday party?
¿Qué día es tu fiesta de cumpleaños? is the required answer. Qué and cuál are listed in your drop-down menu of possible answers. Qué is the only answer to this question that an automaton would accept. Quál (which) would be equally accepted by a live Spanish speaker!
why es and not esta? your birthday is permanent, but the date of your party can change.
Dates use ser. The "permanent/temporary" guide only applies to temporary state or condition.