Let's say a team of well equipped, former special forces soliders left the U.S. Army and formed a "militia." Legally, what could some of their missions be, if they were even allowed to operate? Who would authorize that, if anyone? What are the legal bounds of this organization/militia?
There are no laws restricting a number of people forming a group and calling themselves a militia or similar sounding military name. The factors that limit the activities of any militia or similar group are the same that oversee all of us: state and federal laws.
To begin with, any such militia or group may run afoul of laws if they are perceived to be attempting to confuse the public and state/federal governments of their legitimacy and actual command. And, of course, the federal government will not allow an ad-hoc militia group to subvert their own authority.
A self-organized militia of ex-military types could self-assign "missions" of any nature, as long as they did not break applicable state and federal laws for firearms, applicable laws that require permits for assembly and parades in public spaces, laws regarding threats and intimidation while in public, and laws regarding Sedition (Wikipedia), among others.
These "missions" could be patrolling cities and reporting crimes and helping old ladies cross the street, like an armed version of the Guardian Angels (Wikipedia). But more aligned with current events, some possible examples are the Oath Keepers (Wikipedia): they gather at request (or by their own mission directive) and "provide security" (their description) for other groups - during public rallies and demonstrations - that have (typically right wing) political views aligned with their own. They appear in military garb and are heavily armed, but are allowed to assemble and "operate" (their term), as long as they are within the bounds of state and federal laws. (There are instances of laws being "bent" by law enforcement to allow militias to "operate" i.e. Occupation of the Malheur National Wildlife Refuge (Wikipedia); but typically, federal authorities supersede). But a militia could also claim to be non-partisan, or even left wing, such as the Redneck Revolt (Wikipedia).
Such militias organize and authorize their own "missions", much like any group, like the Boy Scouts. The only outside authorizations they may need are legal permits, where required, for their activities, such as assembling and parading in public, among others. If a militia is run as a 501(c) organization (Wikipedia) they are also bound by those financial and governing laws.
The legal bounds of how a militia operates in public (and in some cases, within private spheres) are - like anyone else - the applicable state and federal laws: for firearms (open carry, concealed carry, long guns, automatic weapons, etc.), laws regarding threats and intimidation (since they are usually armed and their modus operandi is intimidation) and assault, as well as any laws regarding assembly and parades, and sedition.
In the Second Amendment context, the phrase "well-regulated militia" only means that the militia is subject to "the imposition of proper discipline and training." D.C. v. Heller, 554 U.S. 570, 597 (2008).
For the most part, this has no bearing on the answers to your other questions. Unless a group is sanctioned by the government, it gains no special powers by styling itself a "militia," whether it is well regulated or not.
A militia's authority to run "missions" is the same as as any other group, whether it's a book club or a mall security team: They can embark on missions that are within the law, and they can't act outside the law.