You ask:
Is the non-physical amenable to research and investigation?
Ceteris paribus, yes. Many non-physical phenomena are investigated. In the exotic cases for we secular physicalists, we have para-psychology or theological research where the supernatural is presumed to exist. Now, that being said, many who are militant against the supernatural will answer no, so in that vein, you'll find those here who will reject it based on their metaphysical framework; but this knee-jerk response is from those not aware of thir metaphysical biases. From a purely philosophical perspective, the non-physical can be investigated.
Even if you reject the supernatural, the formal sciences are largely taken not to be empirical disciplines primarily. Mathematics and logic is not about utterances and media; it's about ideas. A Platonist will certainly endorse the idea that mathematical objects, which are abstract objects can be investigated despite lacking a physical property. And you won't find many logicians who endorse the idea that propositions have some physical dimension. Ideas, concepts, and meanings are taken to be "above or over" the physical media that conveys them by many, and this bleeds into realist metaphysical positions.
You state:
By methodologies of investigation or research, I mean any procedures that could be followed by an individual to acquire knowledge or gain a deeper understanding of non-physical aspects of reality that the methodology is designed to explore or study.
Here, I'm concerned you are taking a woo-like approach to reality. There is physical reality built on causal closure and spatiotemporal extension, and putative abstractions and non-physical entities are attributed to it in various ways by various thinkers. In modern science, the chief methodology for investigating that without a physical basis is operationalization. Philosophically, the primary method is scientific in that is a form of scientific reduction (SEP). From the article:
Reductivists are generally realists about the reduced phenomena and their views are in that respect conservative. They are committed to the reality of the reducing base and thus to the reality of whatever reduces to that base. If thoughts reduce to brain states and if these brain states are real, then so too are thoughts. Though conservative realism is the norm, some reductionists take a more anti-realist view. In such cases the reducing phenomena are taken to replace the prior phenomena which are in turn eliminated. The idea of mental illness as a type of psycho-neural disorder replaced the idea of demon possession. Demons and their voices have no role or reality in the new theory. The oxygen theory of combustion replaced the phlogiston theory and phlogiston was eliminated.
So, when dealing with things that are outside the bounds of the physical, by setting up repeatable and measurable procedures helps us to determine theories to account for things that don't have directly obtainable empirical properties. Energy is a perfect example. Most physicists will tell you energy is physical, but something like potential energy doesn't actually have directly observable physical properties. This undermines realist arguments and is seized upon by instrumentalists to claim that the case made traditionally by scientific realists is overstated, and that observation and theory are interrelated as in the case of theory-ladenness. There's a gulf between the senses and the intellect created by the phenomenological perspective, and it's not quite so simple to bridge. Just look at the conceptual complexity of Kant's CPR!