What is the ruling regarding women going out in the path of Allah with men and spending the night in the same mosque?

The call to Allah (Da'wah) encompasses both men and women equally. There is no impediment to a woman going out for Da'wah if she is secure from fitnah (temptation/strife) and her going out does not entail any Shari'ah-forbidden act, such as traveling without a mahram (unmarriageable kin) or neglecting an obligation. As for women spending the night with men in the mosque, there is no objection to it if there is a barrier that screens the women; otherwise, it is prohibited due to the potential for forbidden outcomes, such as the exposure of 'awrah (private parts).