Legislatures generally make their own rules.
As I understand it the plaintiff is arguing that the Legislature violated the Constitution by saying this was an emergency, when it wasn't.
The judge is saying that is a matter of Legislative Authority, not a matter of law for a Judge. (i.e. checks and balances)

If the law's constitutionality were questioned he could rule on that, but it wasn't.

He was asked to rule on whether the legislature acted within their constitutional authority.

In NYS, there is a "waiting period" of some sort as I understand it to hold public hearings, print the bill, etc. which can be waived in an "Emergency". The judge is saying that it's not his job to decide whether or not a mass shooting in another state constitutes an "Emergency"/

NYS Supreme Court is a trial court.

The Apellate Division of the Supreme Court is the original Appeals Court... IIRC everyone (except a prosecutor) has a right to 1 appeal to the Apellate Division.

NYS Court of Appeals is the Highest Court, but much like SCOTUS, they decide what they hear and what they don't

In any case, the law (and order to show cause) wasn't related to a gun ban... it was related to how the gun ban was passed, which they contend violate's NYS constitution.

This was much to do about nothing as NYS does not have a RKBA enumerated in their constitution as many states do, and the Constitutionality of the law itself was not questioned.

A constitutional challenge to the LAW would be a challenge via the 2nd Amendment to the Federal Constitution in Federal Court.
This has not happened.

A state supreme court is what other places call a county court -- they're elected by the local county populations and handle things like divorce, felonies, etc. From there, an appeal goes to the 13 appellate districts, then the four appellate divisions, and then the CoA.

NY likes to keep us on our toes .

Forgot about the appellate districts!
But county and town courts are also separate.

I forgot to mention villages can have their own courts inside of towns (which often also have their own courts). Cities have their own courts completely separate from county courts (and on the same level... they're both considered state supreme courts, actually, and unlike village and town courts, usually hear their own felony cases...)

There's a reason I'm not a lawyer: I'd get lost figuring which one of the 60-some supreme courts and hundreds of town, village and city courts I would go to on a given day, haha.