What is permissible for the husband to do as far as having sex with his wife is concerned?
The couple are free to enjoy each other sexually provided that they don`t do it during menses. It is also forbidden for the husband to enjoy the wife from her anus.
Is it permissible to swear by (Allah "The Living")?
Yes, it is permissible because "The Living" is one of Allah`s Attributes.
Is it permissible for a wife to leave her house without asking her husband?
It is impermissible for the wife to leave her house without asking her husband except for a sound reason.
I am a woman, and my natural menstrual cycle lasts six days. In the evening, I perform the testimony (Tashahhud) for myself, and then I engage in sexual intercourse with my husband. After intercourse, I experience bleeding, either immediately or the next day. Is my intercourse with my husband during this time forbidden?
It seems that your menstrual cycle lasts seven days, not six. Therefore, you should wait until the bleeding stops, then perform ghusl and pray. And Allah Knows Best.