The job is the equivalent of a U.S. Attorney, but is not subject to day-to-day supervision by the Justice Department. The special counsel has to be named from outside government and has broad leeway in staffing any investigation.
As special counsel, Mueller technically works for the Justice Department, so the Attorney General has the authority to fire him but only for "misconduct, dereliction of duty, incapacity, conflict of interest, or for other good cause, including violation of Departmental policies."
But because Attorney General Jeff Sessions has recused himself from the case, that authority falls to Deputy Attorney General Rod Rosenstein, who appointed Mueller.
Congress has the authority to create an independent commission to conduct an investigation, much as it did after the assassination of President Kennedy and the 9/11 attacks.
That would require a special act of Congress, the appointment of commissioners from both parties and the formation of an independent staff of investigators. The commission could be given subpoena powers, as Congress did for the 9/11 commission.
But the result of such a commission's investigation is only a report; it would not have the power to press charges or seek civil penalties.
Given the current GOP control of both houses of Congress, the creation of an independent commission is unlikely.
Congress can set up committees for specific investigations, but at this point there are already separate Congressional probes underway, the primary ones being handled by the Senate Intelligence Committee and House Intelligence Committee.