Is it permissible for a friend to take a portion of a debt (13,000 UAE dirhams) in exchange for the value of gifts she previously gave (6,500 dirhams), and does the remaining amount still constitute a debt upon her?

It is not permissible for your friend to take back her gift, as it is authentically reported from the Prophet (peace and blessings of Allah be upon him): "The one who takes back his gift is like the one who takes back his vomit." This is the view of the majority of scholars. If there was an exchange of gifts, the giver cannot, by consensus, take back the gift. However, the Malikis hold that a giver who has not received compensation for his gift has the right to reclaim it as long as it has not perished. If it has perished, he is entitled to its value. The giver's intention for reward is determined by circumstantial evidence. Therefore, if your friend did not receive compensation for her gifts, she may take them back or their value if they were given with the intention of reward.