Is it obligatory to lower the gaze from a picture of a woman's shadow where her hair appears?

Looking at the image you mentioned is not forbidden, as long as you do not fear temptation or lust from it. Jurists have explicitly stated the permissibility of looking at the image of the awrah (private parts) in a mirror and the like, as mentioned in Tuhfat al-Muhtaj. This is supported by the fact that divorcing contingent upon sight does not occur by seeing a reflection in a mirror. The questioner is advised to turn away from obsessive thoughts and stop preoccupying themselves with them.