You can appeal a decision from the highest court of a state to SCOTUS on constitutional grounds. But it has to be federal - a state high court is the end of the road for state law issues.
Yes - Habeas Corpus could eventually get to the SCOTUS. After State Court appeals are exhausted, a writ is filed in Federal District Court, appealed to the Circuit Court, then to the Supreme Court. And I’m a little rusty (not a constitutional attorney) but I think it is only for sentences of incarceration (the Latin means “produce the body”).
Finally, no, a person suing a state does not get you to the Supreme Court. That happens ALL THE TIME. New York has its own court set up for just that purpose, the Court of Claims. Only States suing each other have original jurisdiction before the Supreme Court, which is quite rare.
The actual text of Article III says, In all cases affecting ambassadors, other public ministers and consuls, and those in which a state shall be party, the Supreme Court shall have original jurisdiction.”
Why would that not apply to Trump suing the State of NY for violating his 14th Amendment right to due process?