No. The arrest is not likely to be reported, but if you were fingerprinted, a copy of them will end up in the FBI database.
The FBI does arrest individuals. They arrest people charged with federal crimes. They have the choice to question the individual in a local police department or at their own offices if that is a viable choice.
They are not really comparable. The CIA does not enforce any laws, but collects intelligence overseas. There are no circumstances where the CIA can legally arrest you (besides the inspector general arresting CIA employees for, say, stealing money from the CIA). The FBI can arrest people in the US for violating federal laws, and collects intelligence inside the US.
They can't just arrest you for questioning. What they can do is to invite you in for question.
Yes, the FBI has a program called the FBI Rewards for Justice program that offers monetary rewards for information leading to the arrest or conviction of wanted terrorists, fugitives, or individuals involved in certain criminal activities. The amount of the reward varies depending on the case.
Usually includes routine FBI.
yes
yes
they stake out in front of the persons house and see if they are really selling drugs and if they are they go into the house and arrest them
Hell yeah. They can arrest the president if the circumstances are legitimate.
The job of FBI agents is to investigate actual or suspected violations of federal law, and to arrest criminals whom they identify in the course of their investigations.
Tradk and Arrest Criminals, investigate cyber crimes, fraud, kidnapping, murder.