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 will summarize the answer to your questions in the following points:
1- It is an obligation to pay Zakah on money if it reaches the Nisaab (the minimum amount liable for Zakah) and a whole lunar year elapses on it. If this money is owned by the husband, he is obliged to pay Zakah on it, as regards the wife, if she has no money, she is not obliged to pay Zakah in principle. For more benefit, please refer to Fatwa 93584.
2- If the savings which a woman has reach the Nisaab by itself or with what is added to it of what she owns of other money or tradable items, and a whole lunar year elapses on it, then it is an obligation on her to pay Zakah on this money. The fact that she does not have another constant source of income does not wave her from the obligation of paying the Zakah on that money.
3- It is permissible to transfer the Zakah money to another country if there is a preponderant need that requires this provided that one surely knows or predominantly thinks that in the country where the Zakah must be paid there is no one who needs it and does not find what covers his need from the money of Zakah.
4- It is permissible for a Muslim to pay off his Zakah by himself or to deputize another trustworthy person to distribute it, whether this is an individual or a charitable institution or else, and this is sufficient for him. For more benefit, please refer to Fatwa 88677.
Allaah Knows best.