Is it permissible for a woman wearing the niqab to go out without her husband to public places such as the market and restaurants? And can she travel without a mahram, knowing that the number of women on the plane and at the airport exceeds four?

There is no harm for a woman to uncover her face in a place safe from non-mahram men. It is permissible for her to go out with her mother to the market and similar places with her husband's permission, or if it is known that he does not object. A mahram is required for travel by any means of transportation, and it is not permissible for a woman to travel without one, except for obligatory travel like Hajj and Umrah if she cannot find a mahram and is with a trusted group. It should also be noted that a married woman is not permitted to travel without her husband's permission, even if she has a mahram with her.