Is it permissible to give a sister a gold necklace from the inheritance as part of her share in the price of a building that was purchased from the heirs, or must the necklace be sold first and then her share given to her in cash? And is it permissible to lend her money to buy the necklace and then reclaim it from her as part of her share?

You have two options: either you give her her share of the apartment's value, and then if she wishes, she can buy the aforementioned contract from you. Or, you give her this contract as payment for your debt, under two conditions: First, that it be at the price of gold on the day of payment, so that there is no profit in what is not guaranteed. The Prophet, peace and blessings be upon him, forbade profit from what is not guaranteed. Second, that you do not separate while there is still something between you, so that the required قبض (taking possession) occurs, which prevents riba (interest). The evidence for this is the hadith of Abdullah ibn Umar, may Allah be pleased with them both: "There is no harm in taking them at the price of their day, as long as you do not separate while there is still something between you." We also note that your maternal aunts do not inherit from your mother due to your presence and the presence of your brother.