What is the ruling on someone who makes a mistake in reciting a verse during prayer: must he go back to correct it, should he alert the worshippers after the prayer, and what is the correct intention for the Imam's recitation of the Quran?

It is not permissible to recite the Quran with doubt about what is being recited; rather, one must stop until the correctness of what is being recited becomes clear.

The prayer is valid whether the mistake is corrected or not, but one must repent from reciting with doubt and not return to it.

It is not necessary to inform the worshippers of the mistake because their prayer is valid. The intention to lead (as Imam) is not a condition but is recommended.

What is required is the intention to draw closer to Allah through worship and to achieve humility. The awe you feel might be due to lack of habituation or a pathological fear that has a cure.