Is it permissible to supplicate with what Abraham (peace be upon him) supplicated to his Lord for his heart to be assured, and is this exclusive to prophets only, and is it obligatory to fulfill a vow to fast a month for Allah in exchange for seeing a sign or proof, even if the supplication is not answered?

It is permissible to supplicate to Allah to show you a sign (Ayah) to increase faith and strengthen certainty, as Ibrahim (peace be upon him) did. There is no harm in seeking that which increases certainty. If you vow to fast for a month for the sake of Allah if you see a sign, you are not obligated to fulfill the vow until the condition is met. There is no specific wording for the supplication; you can say: "O Allah, show me such and such a sign."