The US is a common law country, which means law comes from a few sources: statutes (bills passed by Congress), administrative law (rules passed by federal agencies), and common law (rules derived from judgments in cases).
The US Congress passes bills (though at a somewhat decreasing rate in recent years). For example, in 2011, Congress passed the Leahy-Smith America Invents Act to revamp the US patents system. There are thousands of bills enacted in total, around a couple of hundred per year.
The United States Code is a compilation of enacted statutes, which are reordered into titles based on subject area. For example, title 35 relates to patents. Each USC title can include parts of many different statutes, and each statute may go into a few different titles.
Statutes in the USC are law, and will be applied by the courts.
Various federal agencies are empowered to establish regulations to administer their area. Notably, the USPTO is empowered to establish regulations for how patents are applied for. These are published in the Federal Register. There are thousands of these regulations.
The Code of Federal Regulations is a compilation of regulations from the different federal agencies, which are reordered into titles based on subject area. For example, title 37 (which you linked to) relates to patents, trademarks, and copyrights. Each CFR title can include hundreds of different regulations, and each set of regulations may go into a few different titles.
Regulations in the CFR are law (albeit secondary to the statutes in the USC), and will be applied by the courts.
When a case comes before a US court, the court will issue a judgment with their reasoning for how they decided the case. The doctrine established by that reasoning becomes part of the common law. For example, there was a series of cases over the years, which established the doctrine of nonstatutory double patenting. This doctrine does not appear in the USC or CFR, since it does not originate from a statute or regulations, but it is law nonetheless.
This continues to be part of the law until a statute is passed (which renders the original common law doctrine moot) or a higher court overrules it. For example, if the Court of Appeals of the Federal Circuit rules one way, this is law until the US Supreme Court rules in a contrary way.
Case law is not officially compiled in any real way, and practitioners simply have to wade through thousands of judgments to distill what the common law is. There are various secondary sources which attempt to compile doctrines, like the Restatements of the Law, but these are just for convenience and do not themselves have force.
Case law is law, and will be applied by the courts.
The MPEP is a manual for examiners to apply the law. This is because there is so much statute, regulation, and case law that examiners cannot be expected to have read and understood it all, and to consistently come to the same interpretation. The MPEP therefore works as an interface between examiners and the law.
Examiners are required to apply the MPEP (since they cannot act arbitrarily), so applicants and practioners should therefore also follow it, to guide their interactions with examiners. However, it is not in itself law.
The MPEP gets updated when there are changes in the law.
Other agencies can have similar manuals, generally to regulate the operation of their employees. But there is no real obligation to do so.