How can one deal with a wife who is sensitive to her husband's opinions and requests concerning increased religious commitment, and is this considered part of a wife's obedience to her husband, especially regarding leaving the house or going to a female doctor?

The wife is obligated to obey her husband concerning matters of her enjoyment, such as her purification and cleanliness, and what he commands her regarding the performance of obligations, like wearing the hijab (definition) and avoiding forbidden acts. There is no obedience to a creature in disobedience to the Creator. In summary, what the wife must obey her husband in boils down to two matters: obeying him in everything related to the marriage contract and its particulars, and obeying him in refraining from forbidden acts and performing obligatory ones. She is not permitted to leave his house except with his permission, nor to seek medical treatment from a male doctor unless a female doctor is unavailable. The husband should be gentle with his wife when commanding or forbidding her.