Is wearing the khimar desirable, and if not, why? Is it a condition to wear black for the jilbab along with the khimar, knowing that society considers women who wear the hijab to be backward?

A woman's wearing of the khimar (head covering) is an obligation according to Islamic law, and it is not permissible for her to remove it under the pretext of it being considered backwardness or regression by the society she lives in. Indeed, considering the hijab as backwardness is an act of major disbelief (kufr akbar). It is obligatory to wear the khimar that covers the body, and there is no option in this regard. There is no specific color required for the khimar.