Sure.
The FBI is authorized to investigate any criminal offfense embodied in the US Code.
Broad answer is... yes, they could.
The FBI can investigate any federal crime, whether it is considered to be "white" or "blue" collar. If someone bashes the head of a federal officer with a lead pipe, the FBI has the authority to investigate that federal crime.
yes
To stop and/or investigate big crimes like a bank robbery or a murder
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.
The FBI usually investigates a particular place based on reports and calls from the public or other law enforcement agencies.
Tradk and Arrest Criminals, investigate cyber crimes, fraud, kidnapping, murder.
There are a lot of "It Depends" possibilities here, but if the crime crosses state lines in the US, the FBI will be involved.
That is a very complex question. Simply put: the role of the FBI is to investigate the violation of certain Federal Laws and bring those who violated those Federal laws to justice.
They are technically police expect they don't protect the crime scene they investigate
No. The Mormons are peaceful people. Why would they need to be investigated? It's like asking if we should investigate Christianity or Judaism or something.