NOMA (non-overlapping magisteria) is a fallacy. In fact science is concerned to understand all of the natural world, and everything that affects it, even if this interference is supernatural. That is, the science is concerned with the supernatural, provided this is a supernatural interfering in nature or in its understanding.

For science existence is the exception, and inexistence is the rule. That's why phlogiston, Vulcan Planet, luminiferous Ether are all inexistent for science, althout in the past they were all sciencific hypothesis.

for science, if the existence of something new was hypothesized to explain something, but later a new explanation made this  existence unecessary, that new existence is discarded as false (existence is the exception, inexistence is the rule).

God of the gaps?

If the God hypotesis somehow explains or interferes in the natural world, so science will never ever be agnostic regarding this hypotesis.

All that explains or interferes with nature is in the interest of science. And you can prove the nonexistence of something that was hypothesized simply showing that there is already a better explanation that makes the existence of that something unnecessary. Read about phlogiston, the luminiferous ether.

Science has disproved God as well as already disproved the luminiferous ether, phlogiston, the planet Vulcan, the atom Thomson, ultra-mundane corpuscles, vis viva, elan vital, calorie fluid, miasma, ciclol, and many others fail hypotheses.

Your idea of "proof" seems to be coming up with a "better" explanation that makes the existence of something unnecessary.  Of course, "better" is subjective.

I'll help you.  One definition of "better" is that used by Occam's Razor which says the simplest possible explanation is preferred _for the reasons stated below_.

> In science, Occam's razor is used as a **heuristic** technique (discovery tool) to guide scientists in the development of theoretical models, rather than as an arbiter between published models.[1][2] In the scientific method, Occam's razor is **not considered an irrefutable principle of logic or a scientific result**; the preference for simplicity in the scientific method is based on the falsifiability criterion. For each accepted explanation of a phenomenon, there may be an extremely large, perhaps even incomprehensible, number of possible and more complex alternatives, because one can always burden failing explanations with ad hoc hypotheses to prevent them from being falsified; therefore, **simpler theories are preferable to more complex ones because they are more testable**. - Wikipedia

But I'm an engineer and I specialize in making things more complicated than they need to be.
One need not assume that the Designer of the universe feels any need to simplify it down to your current level of understanding.

http://wonderopolis.org/wp-content/uploads//2014/05/Rubenvent1.jpg

Since we can't possibly know what the Technical Requirements are for the Universe, we are unable to state whether the most simple or "natural" explanation meets all the requirements or whether it has been over simplified due to our lack of knowledge of the true "specs".

Therefore, you are stuck with only being able to say "I have found an explanation I like "better" for nature _as I currently understand it_.

This, of course, proves nothing about the actual state of reality at all.