Is it obligatory to change the name "Yazid" for a newborn, knowing that naming with it is disliked by Shias, and was Yazid ibn Muawiyah on the right or on the wrong?

There is no harm in naming someone "Yazid." Some Companions were named Yazid, such as Yazid ibn Abi Sufyan, as were a number of Sunni scholars and Hadith narrators, like Yazid ibn Abi Habib and Yazid ibn Zura'i. No scholar is known to have criticized this naming. As for cursing Yazid ibn Mu'awiyah and faulting him for killing Al-Husayn and fighting the people of Medina, please refer to fatwas number 50285 and 32069.