Is it permissible to professionalize henna artistry as a profit-generating profession, and what is the ruling on engraving specifically for those who wear socks when going out and do not care to cover their feet at home or when receiving visitors?

There is no harm in a woman dyeing her hands and feet with henna, as it is a form of adornment that must be concealed from non-mahram (definition) men, due to the generality of Allah Almighty's saying: "And not display their adornment." Working in the profession of henna artistry for women is permissible, and the money earned from it is lawful, unless the artist knows that the woman will display it in front of non-mahram men. In that case, it is not permissible for her to assist in that, and she should advise and guide her.