Is performing Umrah an obligation for who is capable. Is he accountable for not performing it in this case?
Umrah is an obligation once in a lifetime for every Muslim. When physical and financial ability is present, it is preferable to perform Umrah, as well as Hajj. If one can afford to perform Umrah but not Hajj, then they should begin with Umrah, especially since Hajj nowadays is not accessible to everyone who wishes to perform it. In contrast, Umrah is more open and easier to undertake. And Allah knows best.
What is the ruling on having slight discharges of menstrual blood during daytime of Ramadan, and before Maghrib time?
A woman who is certain that her menses have started during daytime of Ramadan is in a state of menstruation, and her fasting is considered broken at first sight of blood. However, she is rewarded for not breaking fast at the beginning of daytime.
I made a vow "If the Israeli troops retreated out of Gaza strip I will buy a cake from a certain bakery" and it happened. Am I obliged to buy from the same bakery?
You are obliged to buy from the same bakery. And Allah Knows Best.
Is it permissible for a pious Muslim woman to stay with her husband who had quit praying out of laziness?
Quitting prayer is one of the major grave sins that come after disbelieving in Allah, but the wife of such a person isn`t considered divorced, rather, she should exert all her efforts in order to bring him back to the way of Islam. However, if his sin is likely to lure her from the way of Islam, then she had better separate from him by lawful means such as Mokhal`aa (When a wife pays a compensation for her husband in return for divorcing her). Moreover, if she exercises patience, and remains steadfast on her faith, then there is no harm in doing that.