What kind of authority does POTUS has over FBI? Can POTUS directly order DOJ & FBI to close an investigation? After the Watergate scandal, did Congress pass any laws to make FBI a little bit more free when it comes to investigating politicians who are currently in power, because closing investigations based on POTUS recommendation will be a blatant obstruction of justice (which is exactly what happened in Watergate scandal)? Why not make FBI a completely independent investigative organization?
Can POTUS directly order DOJ & FBI to close an investigation?
He can fire the Attorney General and the Director of the FBI, and he has actually done both of those things (to Sally Yates and James Comey respectively, though the former was the Acting Attorney General at the time). Whether he can lawfully "order" the closure of an investigation is unclear and probably very fact-specific, when you take into account laws against the obstruction of justice. If the President reasonably believed an investigation to be entirely without merit, or a waste of federal resources, stopping it would likely be a lawful exercise of his authority. If the President believed that an investigation would harm himself or someone he cared about, and ordered it terminated on that basis, that would most likely be an illegal act.
Regardless of the law, he can certainly make his preferences known, and threaten to fire or actually fire officials who refuse to comply. After that point, deciding whether he violated the law is Congress's problem under the Impeachment Clause.
Why not make FBI a completely independent investigative organization?
Article II of the US Constitution opens with this line:
The executive Power shall be vested in a President of the United States of America.
(Capitalization in original for this and all other quotes from the Constitution.)
In Section 2, it goes on to state that:
... and [the President] shall nominate, and by and with the Advice and Consent of the Senate, shall appoint Ambassadors, other public Ministers and Consuls, Judges of the supreme Court, and all other Officers of the United States, whose Appointments are not herein otherwise provided for, and which shall be established by Law: but the Congress may by Law vest the Appointment of such inferior Officers, as they think proper, in the President alone, in the Courts of Law, or in the Heads of Departments.
Article II is silent on the question of whether the President may remove such officers. However, the Supreme Court has repeatedly held, most recently in Bowsher v. Synar (1986), that executive branch officials serve at the pleasure of the president and may be unilaterally removed. Humphrey's Executor v. United States (1935) did hold that officers of regulatory agencies, such as the FTC, are to be treated differently, but that's because of their quasi-legislative roles rather than because Congress can generally limit the President's firing authority.
But that's actually not the whole story. Under Morrison v. Olson (1988), Congress is allowed to establish a separate independent counsel, with very little executive branch oversight.* This is on the theory that:
The [Independent Counsel] Act, taken as a whole, does not violate the principle of separation of powers by unduly interfering with the Executive Branch's role. This case does not involve an attempt by Congress to increase its own powers at the expense of the Executive Branch. The Act does empower certain Members of Congress to request the Attorney General to apply for the appointment of an independent counsel, but the Attorney General has no duty to comply with the request, although he must respond within a certain time limit. Other than that, Congress' role under the Act is limited to receiving reports or other information and to oversight of the independent counsel's activities, functions that have been recognized generally as being incidental to the legislative function of Congress. Similarly, the Act does not work any judicial usurpation of properly executive functions. Nor does the Act impermissibly undermine the powers of the Executive Branch, or disrupt the proper balance between the coordinate branches by preventing the Executive Branch from accomplishing its constitutionally assigned functions. Even though counsel is to some degree "independent" and free from Executive Branch supervision to a greater extent than other federal prosecutors, the Act gives the Executive Branch sufficient control over the independent counsel to ensure that the President is able to perform his constitutionally assigned duties.
Based on a careful reading of the bolded sections, it seems clear to me that the Supreme Court would be far more skeptical of an independent FBI than an independent special counsel. This is because the FBI lies at the heart of the executive branch's Constitutional responsibility to "take Care that the Laws be faithfully executed". Since the FBI is the primary federal law enforcement agency, limiting the president's power over it would surely abrogate his ability to execute the law.
* It should be noted that Robert Mueller is not in this bucket; he can be removed by the Attorney General, or by the Deputy Attorney General as long as the Attorney General remains recused from the investigation. The Independent Counsel Act discussed in this ruling expired in 1999.