Dtyjlui

It is my understanding that the state and national government of the United States are separate entities (that's what federalism is, combining smaller states into a larger one).

Could unhappy citizens overthrow a state government while still remaining on favorable terms with the national government?

Could state militaries overthrow their own state government and establish a new state government and still be on good terms with the national government?

Has this been done in the past?

Has this happened in other federations around the world?

Can state governments be overthrown without national government involvement?

Yes. It happens sometimes when there's an election.

Could state militaries over throw their own state government and establish a new state government and still be on good terms with the national government?

Most likely no. Each state also has a constitution, and prescribes how changing the makeup of each state government should occur. Should a state's national guard (or other state-level militia) attempt a state level "coup," the Supreme Court of the state would most likely rule that such an action violated the state's constitution. In all likelihood, the action from the perspective of the federal government I believe would be viewed not much differently as general insurrection and would be responded to.

Of course, each state is different and has different constitutions. For example, New Hampshire's contains the Right of Revolution:

Government being instituted for the common benefit, protection, and security, of the whole community, and not for the private interest or emolument of any one man, family, or class of men; therefore, whenever the ends of government are perverted, and public liberty manifestly endangered, and all other means of redress are ineffectual, the people may, and of right ought to reform the old, or establish a new government. The doctrine of nonresistance against arbitrary power, and oppression, is absurd, slavish, and destructive of the good and happiness of mankind.

Since this is part of the state constitution, the state Supreme Court may be swayed by the arguments of the state militia depending on the exact circumstances. Other state constitutions contain similar clauses, a good listing of states with a positive right of revolution can be found on Wikipedia.

If the state Supreme Court isn't swayed, the federal Supreme Court might, but the wording presents a bit of a conundrum since if we're appealing to the courts then by definition that means that redress is available. This redress also becomes available every time there is an election. As long as the people have faith in their state level elections, then the choice of the make-up of a government still belongs to the people.

See also Article 4, Clause 1 of the U.S. Constitution:

The United States shall guarantee to every State in this Union a Republican Form of Government [...]

A military overthrowing a popularly elected government is not a republican form of government.

Has this been done in the past?

Not in the manner are describing, but the legislatures of many states attempted to dissolve the union between them during the Civil War, which was later ruled unconstitutional by the Supreme Court. I doubt a state-level militia would get favorable treatment for attempting to do something similar.

Thanks to @Drunk Cynic and @guest271314 , I think I can cobble together an answer to my question.

@guest271314 pointed out the State of Jefferson debate, which led me to find this thread: Why can't the counties that support the proposed state of Jefferson appeal directly to the federal government for statehood?

The answer given in this thread mentions

Article IV, Section 3, Clause 1 of the US Constitution:

New States may be admitted by the Congress into this Union; but no new
States shall be formed or erected within the Jurisdiction of any other
State; nor any State be formed by the Junction of two or more States,
or parts of States, without the Consent of the Legislatures of the
States concerned as well as of the Congress.

This shows that probably any means of "over throwing" (legal or forcefully) state government is not going to be able to join the United States because the new state occupies the space occupied by the already recognized previous state government.

@guest271314 provided an example of this occuring in the past around the world:

 Republic and Canton of the Jura in Switzerland.

After World War II, a separatist movement campaigned for a secession
of Jura from the canton of Bern. After a long and partly militant
struggle, which included some arson attacks by a youth organisation
Les Béliers, a constitution was accepted in 1977. In 1978 the split
was made official when the Swiss people voted in favour, and in 1979
the Jura joined the Swiss Confederation as a full member.

Perhaps this replacement of state governments can occur in other federations around the world, but for the United States it does not seem possible.

Although this does not fully answer the question, I just wanted to point out that one of the driving forces behind the current US Constitution was Shays' Rebellion, am armed uprising in Massachusetts that the Confederation government helped suppress. The Rebellion was nearly successful, and many feared a subsequent rebellion would overpower state militias. Thus, there was a perceived need to create a national government strong enough to protect the states from further internal insurrections. See Joseph Parker Warren, The Confederation and the Shays Rebellion, 11 Am. Hist. Rev. 42 (1905).

For example, in Federalist 43, Madison references the Rebellion ("[a] recent and well-known event among ourselves") and describes how a federal power to suppress insurrection within the states would aid democracy:

Why may not illicit combinations, for purposes of violence, be formed as well by a majority of a State, especially a small State as by a majority of a county, or a district of the same State; and if the authority of the State ought, in the latter case, to protect the local magistracy, ought not the federal authority, in the former, to support the State authority? Besides, there are certain parts of the State constitutions which are so interwoven with the federal Constitution, that a violent blow cannot be given to the one without communicating the wound to the other.

At the time, of course, part of this fear was fueled by the effect that much of the population was then enslaved, and the southern states were worried about the possibility of a slave revolt. See Pauline Maier, The People Debate the Constitution, 1787-1788 (Simon & Schuster 2010), at page 274. Madison makes note of this as well:

I take no notice of an unhappy species of population abounding in some of the States, who, during the calm of regular government, are sunk below the level of men; but who, in the tempestuous scenes of civil violence, may emerge into the human character, and give a superiority of strength to any party with which they may associate themselves.

Although this reasoning was ultimately used by Lincoln in the Emancipation Proclamation, at the time, Madison was suggesting that the federal government would help southern states suppress slave revolts, thus giving the South a reason to ratify.

To this Madison adds:

In cases where it may be doubtful on which side justice lies, what better umpires could be desired by two violent factions, flying to arms, and tearing a State to pieces, than the representatives of confederate States, not heated by the local flame?

In other words, national power will protect states and make the country as a whole more stable.

I cannot say how the current administration would actually react to an attempt to overthrow, e.g., New Jersey, but it would be keeping with the spirit of the current Constitution for the federal government to help New Jersey in suppressing the rebellion.

Could unhappy citizens overthrow a state government while still
remaining on favorable terms with the national government?

Could state militaries over throw their own state government and
establish a new state government and still be on good terms with the
national government?

If read the question correctly, one example is the Republic and Canton of the Jura in Switzerland.

After World War II, a separatist movement campaigned for a secession
of Jura from the canton of Bern. After a long and partly militant
struggle, which included some arson attacks by a youth organisation
Les Béliers, a constitution was accepted in 1977. In 1978 the split
was made official when the Swiss people voted in favour, and in 1979
the Jura joined the Swiss Confederation as a full member.