Kdjykuj

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).

State governments seem to be much smaller than national governments, and they also affect the lives of it's citizens more than the national government does. 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?

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.

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.