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Summarized Fatawaa

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.

What is the ruling on someone who perform their prayer after finishing it because they believe they missed a Rak'a, a prostration, or that they did not perform it correctly (thinking their prayer was invalid)?

If he was sure that his prayer is void then reperforming it is an obligation along with figuring out the reason of invalidity so long as this wasn't out of uncertainty. And Allah Knows Best. 
 

Does an internal examination require ghusl?

Ghusl isn't required for making internal examination; however, it invalidates one's fasting. And Allah Knows Best.

I took a car from the Orphans' Fund Development Institution; is there usury (Riba) involved in this transaction?

If the Sharia's conditions were met, then there is no harm in this. And Allah The Exalted Knows Best.