Does the U.S. constitution require that state courts be independent from other branchs of the state government? For example could a state alter it’s constitution so that the legislature could overule it’s supreme court (or even abolish it’s supreme court and transfer it’s fuctions to the legislature)?
The Constitution guaranties each state a “Republican” form of government. What exactly a republican form of government is is left up to Congress by Luther v Borden (1849).
Voting rights are covered by Baker v Carr (1962) guarantying “one person - one vote”, but that’s not what you’re asking. I guess that theoretically a state could set up an absolute monarchy if it wanted to and still be legal, but then a group of people could set up a different government and petition Congress for recognition. Study the formation of West Virginia to see how this happens.
The one thing that you talk about they can’t do is have a monarchy. A monarchy, by definition, is not a “republic” because a republic is a government which governs on the basis that the power to govern resides in the people themselves, not a single person (monarch), or a few people (oligarchs), or the church (theocrats), or even the Clintons (ok ok, I’m sorry, had to be said, and just as easily could have said the Bushes). 
As to the OP, it is perfectly possible to have a republic without separation of powers. There are several around. Functionally, the United Kingdom is one, even though they insist on clinging to the trappings of monarchy. In the UK, the judiciary is not independent of Parliament. Other examples exist as well.
A slightly related thread from a while back: Could a state decide to switch to a parliamentary system?