All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
Divorce does not take place just because the two spouses are staying apart from each other, unless the husband utters divorce. For more benefit, please refer to Fataawa 90233 and 89825.
Moreover, it is not necessary for divorce to be uttered three times so that it takes place, rather if the husband says divorce once to his wife, then it takes place. However, if a husband divorces his wife three times, then this is an irrevocable divorce and he cannot remarry her unless she marries another man. For more benefit, please refer to Fataawa 82541 and 82330.
As regards your wife, if what you said about her is confirmed, then she is a disobedient wife, but you should not hasten to divorce her especially that she asked you to give her a last chance. Therefore, we advise you to undertake all permissible means to reconcile with her; in this regard please refer to Fatwa 92474.
Then, if she becomes righteous and regains her senses, all perfect praise be to Allaah, otherwise, if she insists to continue on what she was doing, then it is more appropriate to divorce her as there is no good for you to keep a wife who does not obey you as a husband, persists in abandoning Hijab and speaks to men. It should be mentioned here that it is not permissible for her to discuss with a non-Mahram man (a Mahram, i.e. a spouse or a person whom one is permanently prohibited from marrying due to suckling, marital, or blood relations) even at work except according to the necessity.
For more benefit on the ruling of a woman's work and her mixing with marriageable men, please refer to Fatwa 82399.
For more benefit on a woman refusing to share the bed with her husband, please refer to Fatwa 81787, and Fatwa 92026 about the obedience of a wife to her husband.
If we presume that divorce takes place and you are financially unable to pay the dowry to her, then she is obliged to allow you time until it becomes easy for you to repay. Allaah Says (what means): {And if someone is in hardship, then [let there be] postponement until [a time of] ease.}[Quran 2:280]. However, as soon as it becomes easy for you to repay, you are obliged to do so.
Allaah Knows best.