Is it more virtuous to perform Hajj on behalf of the deceased, given the possibility of correcting errors from a previous Hajj? Does the one performing Hajj on behalf of another receive the same reward? And is a mere intention in the heart sufficient for performing Hajj on behalf of another?

If your Hajj is valid, then you are permitted to perform Hajj on behalf of this man, and you will receive a great reward. Scholars have differed regarding the reward for the proxy: some are of the opinion that he receives the reward for Hajj if he is a volunteer, while others believe that he only receives the reward for the acts of worship he performed, and the reward for Hajj is exclusive to the one on whose behalf it is performed. It is recommended, upon entering into ihram (definition), to vocalize the ritual and the name of the deceased (e.g., "Labbaik, Hajj on behalf of so-and-so"). However, a heartfelt intention is sufficient, even though vocalization is the Sunnah (definition) at the Miqat (definition). As for taking a fee for performing Hajj on behalf of another, the majority of scholars disliked it if the intention was solely for worldly gain. However, if one intends the Hereafter and uses the fee to cover the expenses of Hajj, then there is no harm.