Is it permissible for a Muslim woman to travel by plane for an hour and a half and stay for a month with her sick sister in her home to care for her child, provided she observes complete hijab in front of her sister's husband and avoids being in his presence, and is it permissible for her to include a picture without a head covering in her visa application?

It is not permissible for a woman to travel without a mahram, nor to reside with her sister's husband in the same house if this entails khalwa (seclusion with a non-mahram), uncovering, or displaying adornment. It is also not permissible for her to remove her khimar (head covering) in front of non-mahram men. One should seek a woman to help the sister, who is a mahram to her husband, or send her son to you until her treatment is concluded. Necessities are to be estimated according to their true measure.