"Did you phone me?" is also perfectly good English and should be accepted.
Can this mean both phoning someone or calling someone's name vocally?
Phone only. The correct Ukrainian word is телефонувати, ти телефонувала. Дзвонила is a bit russism but also used.
Call by name is кликати, кликала.
Someone said it means just phoning someone. I'm not sure though
The verb "дзвонити", always uses dative?