What is the ruling on engraving henna for non-Muslim women with the intention of trade, and what is the ruling on photographing henna engravings on the palm of the hand and the foot only and posting them online without mentioning the girl's name?

There is no harm in applying henna for non-Muslim women, because it does not entail humiliation for the Muslim. It is permissible for a Muslim to hire himself out to a dhimmī for a specific task, such as tailoring or dyeing, or for a non-service job for a known period, because it does not involve humiliation of the Muslim. Ibn Rushd mentioned that a Muslim hiring himself out to a Christian or a Jew falls into four categories: permissible for work within the Muslim's home, disliked if he takes over all of his work, forbidden if he is under his direct authority as a servant, and haram if the work is impermissible, such as making wine. The jurists have agreed on the permissibility of a disbeliever serving a Muslim, and the permissibility of a Muslim hiring himself out to a disbeliever for a specific task.

As for displaying images of the mentioned body parts, it is not permissible if there is a risk of fitna. However, if fitna is not feared, we hope there is no harm in it, and some scholars have permitted it.