Will a person be rewarded for studying engineering and English, and working in this field, if they adhere to Islamic guidelines, with the aim of excelling and presenting a good image of Islam to the West, knowing that it involves mixing [between genders]?

It is most befitting for a woman to remain in her home. If she needs to work, she should choose what suits her nature and femininity. Engineering work may not be suitable for her physically or due to the issue of intermingling (free mixing). Teaching male students involves concerns about unregulated intermingling. She can work in scientific research in telecommunications or teach women. And whoever gives up something for the sake of Allah, Allah will compensate him with something better.