The preposition את is not necessary here? Or not allowed?
I would expect it, but maybe it's a poetic thing
Allowed, but not required.
נדמה לי צריך להיות התשמע את קולי
Is " Will you hear my voice" in Hebrew, the same as "Do you hear my voice"?
No, because this sentence uses the future tense תשמע tishma'; if we want to say "Do you hear" it's just אתה שומע ata shoméa. Female equivalent would be תשמעי (tishme'i, will you hear) and את שומעת (at shoma'at, do you hear) respectively.