Asking about agencies such as FDA, FCC, and many others under the US Executive Branch.
When any of these agencies decides to change regulations,such as the FDA banning a previously allowed drug, do they need Congressional approval?
How about things like spectrum auctions in the case of the FCC. Can they do that on their own?
How much say does Congress have in these agencies?
They have a lot of freedom within the broad legislative outlines passed by Congress, and Congress also has a lot of say. Especially when two different parties control the two, they go back and forth.
Take the recent example of EPA with Carbon pollution (vastly simplified from reality here).
Congress sets the broad outline saying “yea, verily, the EPA will regulate pollution”.
EPA regulators look at the definition for pollution and says “by every definition and scientific research, carbon dioxide going into the air is pollution” and starts creating regulations to limit that (just like every other pollutant).
Big coal/oil lobbyists send money to Congress, which then might change their direction to EPA to say “EPA will regulate pollution, but not carbon dioxide” and those rules get stripped from the books.
That is an impossibly broad question to answer, but let me take a stab.
The ability of agencies to do things, the vast majority of cases, depends on legislation that enables the agencies to do those things. (The exceptions to this are pretty rare, but to use one example, the ability of the Department of Defense to employ troops in defense of an attack on this country is a constitutionally derived power, not a statutory power.)
The legislation allowing agencies to do things can be very specific in some cases, and very general in others. The ability to regulate our currency under the Federal Reserve Act is extremely broad, and intentionally so, to distance these monetary decisions from the political process.
On the other end of the spectrum, the ability of the Executive Branch to appoint an independent counsel, such as the one that investigated Nixon and Clinton, depended on a statute that expired quite some time ago. The President can no longer appoint a person to do a job that has no legal basis.
To the best of my knowledge as to your specific questions, I believe the FDA has the ability to revoke approval for a drug at any time, but I’m very hazy as to the requirement for a scientific basis for doing so. And for spectrum auctions, again I’m not quite clear as to the enabling legislation applying to a general power of the FCC to conduct auctions, or that each auction is specifically authorized. I was involved at the very, very outermost fringes of a recent auction, and my recollection was that it was specifically directed by Congress.
The one important caveat to all this is that the Congressional Review Act allows Congress, on an expedited basis, to basically shoot down regulations drafted by the Executive Branch if those regulations go too far in the opinion of the Legislative Branch.
Don’t forget, the system of checks and balances incorporates ALL THREE branches.
Big coal/oil can also sue the EPA in Federal court arguing that the Agency overstepped its legal mandate, and the court can rule in agreement.
That said, it seems the controlling precedent is Chevron v. NRDC (1984), in which the Supreme Court held that ambiguity in the enabling law as to the Agency’s jurisdiction should defer to the Agency. So the courts have less impact than I would have expected, considering how many times you see lawsuits against agency decisions. Go figure. 
Chevron is the controlling case when there’s ambiguity in the federal statute at issue, but that does not occur frequently. Most of the time, the statutory language is fairly unambiguous, in which case courts have a lot of say in the interpretation of the law (also, keep in mind that the question of whether a statute is ambiguous or not, and thus whether Chevron applies, is determined by the Court itself, so clearly the courts have a lot of power in interpreting the statutory and regulatory language).
With respect to the OP’s question, the authority that federal agencies have is largely defined by the Administrative Procedure Act. The APA lays out the procedures in which federal agencies may promulgate regulations published in the Federal Register, and how agency determinations may be subject to judicial review by courts. More directly to the point of the OP’s question – no, agencies do not need Congressional approval when promulgating regulations, but Congress has the authority to change the underlying law in which the specific regulation was based. In practice, however, regulations generally track very closely to the federal statute granting the agency the authority at issue. Some regulations are verbatim copies of the statutory language.
It all depends upon Congress.
Congress can pass an act which establishes a new law, which needs no regulation. For example, Congress could pass a law that says, “No one shall manufacture, produce, or use a cigarette. Violation of this statute is a felony offense punishable by six months in prison.” This requires no regulation to implement, indeed, there’s no indication from Congress that any particular federal agency has the authority to do so.
Congress can pass an act which establishes a new law, and direct a particular agency to pass a specific regulation (or regulations) to help implement the law. For example, “No one shall manufacture, produce, or use a cigarette. The FDA shall within six months of adoption of this law issue a regulation defining what a ‘cigarette’ is. Violation of this statute is a felony punishable by six months in prison.” Under this law, the FDA has a specific mandate to issue a specific regulation, which is to define the meaning of “cigarette.” Notice that Congress has not put limits on that regulation, other than that it must be promulgated no later than six months from passage of the law and signature by the President.
Presumably, under this statute, if the FDA in two years time decided the definition needed to be updated, they could do so. But the FDA would not be able to issue a regulation which defines what “manufacture, produce or use” means.
Congress could pass an act which establishes a new law, but which leaves a federal agency in charge of fleshing out the parameters of the law. For example, Congress could pass the following law: “No one shall manufacture, produce, or use a cigarette. The FDA shall issue regulations governing the implementation of this law, including the meaning of the term ‘cigarette’. Violation of this statute is a felony offense punishable by up to six months in prison, in accordance with the regulatory scheme issued by the FDA.” Under this statute, the FDA has a broad mandate to define all aspects of illegal cigarette manufacture, production and usage, including setting guidelines for determining how offenders will be punished. There are no limits on the regulatory scheme as to timing or content, other than that it must include a definition of “cigarette.” The FDA’s mandate is not explicit as to length of time, but impliedly could be considered indefinite.
There are numerous combinations available. A lot of it has to do with how trusting Congress is of the federal agencies. In the old days, statutes tended to be very limited in detail, allowing federal agencies broad power to implement them. That’s long since ceased to be true; the US Code is almost as complex as the Code of Federal Regulations any more. 
I should also point out that Congress can control federal agencies through the purse strings.
Agencies also have to open up their proposed regulations to a public comments period; the agency has to consider and address these comments when it publishes the final rules or regulations on the matter. Obviously, concerned parties that may be affected by the proposed regulation can raise their concerns with Congress members, so that’s another way agency decisions is ultimately affected by Congressional authority.
Well, Obama couldn’t get the DEA to deschedule marijuana. And I don’t think Congress (or the courts, or voters) had much say in that decision.
Do you have a cite that the President asked/directed the DEA to do so?
No, but he did direct the DOJ to stop prosecuting marijuana crimes in states where it is legal, but the DEA still raided dispensaries in Colorado. Maybe he didn’t ask to deschedule, but he made his wishes clear in this case and the DEA ignored them. I figured their failure to deschedule pot was more of the same, but maybe not.
Obama’s stance was to let states decide on marijuana use. To have the DEA deschedule it would have been a federal de facto decriminalization of the marijuana laws, just what he didn’t want to do because it would have met with massive resistance from anti marijuana states and advocates. Also, the DEA doesn’t follow wishes, they need to be directed to do something by a law passed by congress or a decision by a court. If Obama had directed them to deschedule marijuana there would have been pushback by congress and lawsuits filed in Federal Court for over stepping his authority. The dude was savvy and smooth, I miss him already.