Is a wife obligated to obey her husband who refuses to let her pray in public places, which leads to her missing prayers, and is it permissible for her to perform tayammum for prayer if she cannot find a suitable place for ablution in public places?

Obeying the husband in abandoning prayer until its time has expired is not permissible, because there is no obedience to a creature in disobedience to the Creator.

Prayer must be performed before its time expires. If the husband prevents his wife from doing so, she should not go out with him again.

As for ablution in open restrooms, tayammum (dry ablution) is not sufficient when one is able to use water. There is no harm in exposing the forearms, legs, and hair in front of disbelieving women, because the 'awrah (intimate parts) of a Muslim woman in front of a disbelieving woman is the same as her 'awrah in front of a Muslim woman, and this is the preferred view among the majority of scholars.