Absolutely. If you say one word against the United States Government while on United States soil, not only can you be arrested, but perpetrators are usually denied trial by jury of their peers and given the death penalty.
Hell yeah. They can arrest the president if the circumstances are legitimate.
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.
They can't just arrest you for questioning. What they can do is to invite you in for question.
President Donald Trump was rejected from the FBI before becoming president.
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.
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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
No FBI Director has ever become the US President, nor has any FBI Special Agent become the US President. President Nixon applied to become a Special Agent but was not accepted for this position.
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.