Is it permissible to pray in the outdoor area of a mosque while there is car noise, and does that contradict the reverence for the Qur'an?

There is no harm in praying in the aforementioned area in the mosque courtyard, and this does not contradict showing respect for the Qur’an. What contradicts showing respect for the Qur’an is reciting it aloud in places where people are busy and noisy, such that they cannot listen to it, as this constitutes an indignity.

However, if the noise is loud and disrupts khushu' (devotional humility), it is preferable to seek a quiet place for prayer, as this is more conducive to the presence of the heart. It appears that the idea mentioned is from waswas (whispers/obsessive thoughts), and it is advisable to disregard it.