Does sending gifts to my children's non-Muslim female teachers, in order to increase their interest in them, count as an act of allying with disbelievers?

It is permissible to give a gift to a disbeliever, and this is not considered an act of loyalty to disbelievers. The proof for this is that Umar ibn al-Khattab gifted his polytheist brother a robe. Al-Nawawi mentioned that Umar's hadith indicates the permissibility of a Muslim gifting a disbeliever a garment or other items.

However, if these gifts to disbelieving teachers would lead to favoritism towards your children at the expense of other students, then it is not permissible because it is an injustice, and injustice is forbidden even if it is against a disbeliever. Allah Almighty says: "And let not the hatred of a people prevent you from being just. Be just; that is nearer to righteousness."

Furthermore, a gift given in exchange for a right that must be fulfilled or a wrong action that must be abandoned turns into a bribe, and the Prophet, peace and blessings be upon him, cursed the briber and the bribed.

In addition to that, raising children at the hands of disbelievers is fraught with the dangers of corrupting their beliefs and raising them to imitate disbelievers. It is the duty of a parent to protect their children from everything that brings them closer to the Fire. Allah Almighty says: "O you who have believed, protect yourselves and your families from a Fire."