Is the amount paid by the groom to the bride on the wedding night, in addition to the paid mahr, considered part of the Sharia?

The Mahr (dower) is an exclusive right of the wife that the husband must pay her. He is not religiously obligated to pay anything else, such as what is called "Sadaq Laylat al-Dukhla" (dower of the wedding night) or "Haqq al-Fattashah" (inspection fee), as these are bad customs that violate Islamic law, which encourages facilitating marriage.

Equipping the woman or the marital home is an obligation upon the husband, and it is not obligatory upon the wife or her guardian unless they voluntarily offer to do so.