Is the prayer valid if one vocalizes the intention to pray, such as saying, "I intend to pray Dhuhr prayer, four rak'ahs, facing the Qiblah for the sake of Allah Most High," and if the Adhan is called before every prayer (Sunnah and Fard)?

Your prayer is valid, but your verbalization of the intention is incorrect because it is an innovation (bid'ah). As for the call to prayer (adhan) for the Sunnah prayer, it is not legislated. However, a woman's adhan and iqamah for the obligatory prayer are commendable according to us.