Can the president or the Congress pass a law to dismantle the super-delegate system used by the DNC? Or is it somehow outside their power to do so and each party can set up their own rules as long as it doesn't conflict with existing laws? I am tempted to believe this being a rule within an organization, the state has no say and can't do anything to deprive the DNC of their election system.
Presidents can't pass laws. They can propose laws. They can sign laws once the laws have passed, but they can't, themselves, pass laws. So any question, can the president pass a law to do anything, can be answered no. So no.
Congress can pass laws. In particular, Congress can pass laws regulating how elections are operated. They could take away the ability for party conventions to put candidates on the ballot and replace that primary mechanism with a new mechanism. The parties would then be free to choose their candidates in whatever way they desired. But those candidates would only appear on the ballot if they complied with the other requirements.
The basis for such a law would almost have to be one or more of the fifteenth, nineteenth, and twenty-sixth amendments. Because those are the only parts of the constitution that currently give Congress the power to regulate elections. Since the superdelegate system used by the parties clearly does disempower most voters in favor of the superdelegates, this seems rather straightforward. Another issue is how states limit access to primaries and caucuses. Congress could pass a law regularizing that so as to avoid disenfranchisement.
An example of a system that could be passed would be the top two primary used in California, Washington, and Louisiana. Of course, there would be added difficulty in wording to handle the fact that technically the presidential election chooses electors rather than candidates.
No, not likely. The DNC is a private entity, and is not a part of, nor is it funded by, the federal government. Because of this a federal judge recently dismissed a lawsuit over the internal working of the DNC (PDF):
To the extent Plaintiffs wish to air their general grievances with the DNC or its candidate selection process, their redress is through the ballot box, the DNC's internal workings, or their right of free speech — not through the judiciary.
To the extent Plaintiffs have asserted specific causes of action grounded in specific factual allegations, it is this Court's emphatic duty to measure Plaintiffs' pleadings against existing legal standards. Having done so . . . the Court finds that the named Plaintiffs have not presented a case that is cognizable in federal court.
Furthermore, any improprieties in the operation of the primary selection process (such as discrimination) would be handled by existing law. Additionally, primaries are regulated at the state level, rather than the federal level.
The Constitution dictates the powers enumerated to the President and Congress. It would have to be a state-by-state basis. The Tenth Amendment states:
The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people.
Since there is no mention in the Constitution of the way private organizations like the DNC handle elections, it would be up to the states.