Is paying Zakat through a close person for a worldly gain considered showing off (riya') or seeking reputation (sum'ah)? And must it be paid again? If so, should the same amount be re-paid based on the previous hawl (year), or should a new nisab (minimum threshold) and hawl be calculated?

It is permissible to appoint someone to distribute zakat. If you pay your zakat sincerely for the sake of Allah Almighty through a trusted agent, your zakat is valid even if you experience feelings of showing off (riya') or seeking reputation (sum'ah) after its distribution. The blameworthy riya' is originally intending other than the face of Allah with an act of worship. If a thought of riya' occurs after commencing an act for Allah and is repelled, it does no harm. If one succumbs to the thought of riya', Imam Ahmad was of the opinion that his action is not invalidated thereby. Your zakat is valid, and you are not required to pay it again.