Is it permissible to supplicate in other than Arabic in prayer after the Tashahhud, even with supplications mentioned in the Sunnah? And is it permissible to supplicate with Quranic supplications not mentioned in the Sunnah?

Firstly, if the worshipper is proficient in supplicating in Arabic, it is not permissible for him to do so in another language. However, if he is unable, there is no objection to supplicating in his own language, with the obligation to learn Arabic. As for outside of prayer, it is permissible to supplicate in a language other than Arabic, especially if it leads to greater presence of heart, for Allah knows the intention of the supplicant even if his tongue is not perfectly eloquent.

Secondly, there is no objection to supplicating with Quranic supplications, as they are among the most eloquent. It is also befitting to supplicate with the legitimate supplications mentioned in the Quran and Sunnah (definition).