If the worshipper can say du’aa’ well in Arabic, it is not permissible for him to make du’aa’ in any other language.
But if the worshipper is unable to make du’aa’ in Arabic, there is no reason why he should not make du’aa’ in his own language, so long as he starts learning Arabic in the meantime.
With regard to making du’aa’ in languages other than Arabic outside of prayer, there is nothing wrong with that, especially if that will make the worshipper more focused in his du’aa’.
Shaykh al-Islam Ibn Taymiyah said:
It is permissible to make du’aa’ in Arabic and in languages other than Arabic. Allaah knows the intention of the supplicant and what he wants, no matter what language he speaks, because He hears all the voices in all different languages, asking for all kinds of needs.
Majmoo’ al-Fataawa, 22/488-489.
There is nothing wrong with reciting du’aa’s mentioned in the Qur’aan even if there is no report in the Sunnah that the Prophet (peace and blessings of Allaah be upon him) recited them in his du’aa’. They are all good and contain guidance. Most of the du’aa’s of the Prophets and Messengers that we know are from the Qur’aan. Undoubtedly their du’aa’s are the most eloquent and most profound in meaning.
Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him) said:
People should make du’aa’ by reciting the prescribed du’aa’s that are mentioned in the Qur’aan and Sunnah, because these are undoubtedly virtuous and good, and this is the straight path. The scholars of Islam and the imams have mentioned the du’aa’s that are prescribed in Islam, and turned away from the innovated du’aa’s, so we should follow them in that.
Majmoo’ al-Fataawa’, 1/346, 348.
And Allaah knows best.