The Scientific Method is essentially the effort to discover causation in perceived correlations of collected data. In other words it hunts for 'recipes' that can be applied to consistently achieve a desired result, and as such does not inquire into why something happens, but rather into what makes it happen. Philosophy on the other hand is more concerned with the why: what effect does this have on the trustworthiness of the Scientific Method as an philosophical tool?
It can. The methodologies of experimental philosophy do attempt to apply the methodologies of scientific argument (though the philosophy of science does have quite a lot to say as to what constitutes a fact) to questions of moral philosophy. My own research also attempts to discern a philosophy of data through adoption of scientific techniques though it certainly isn't science.
Experimental philosophy, called x-phi for short, is a new philosophical movement that supplements the traditional tools of analytic philosophy with the scientific methods of cognitive science. So experimental philosophers actually go out and run systematic experiments aimed at understanding how people ordinarily think about the issues at the foundation of the philosophical discussion.
Elements of science can be incorporated extremely well into philosophy, simply because of the unusual techniques providing somewhat unusual insights.
I'm not sure I understand what the question is here...
Essentially, yes. The scientific method can be applied to certain subsets of the general discipline known as "philosophy". It's merely a set of guidelines intended to facilitate a rigorous inquiry. There's nothing particularly unique about its relationship with the natural sciences. Anyone conducting research can apply it.
Here are a couple of other ideas on how it might to relate to what we now call "philosophy":
There are empiricist philosophers who have made an entire career out of applying the scientific method to their work, or at least arguing that it could be applied to their work.
Bertrand Russell is a good one to investigate here, starting with his essay, On Scientific Method In Philosophy, which appears to be available online.
In addition, David Hume's strong empiricist bent could be seen as strongly resembling a somewhat primitive "scientific method", and his ideas on deductive and inductive reasoning were actually quite influential in developing the modern notion of the scientific method.
Also consider the subject of "natural philosophy" (which is the older name for what we now call merely "science") where the scientific method was surely applied rigorously. Before the development of modern science, philosophy and science were seen as one in the same. In fact, many long-established universities still appoint "Chairs of Natural Philosophy", commonly occupied by professors of physical science nowadays.
There is also an entire division of philosophy known as the "philosophy of science" that's concerned with such things as the development of the scientific method. Perhaps they don't approach philosophical inquiry using the steps of the scientific method, but it must certainly play a role!
In fact, the scientific method was itself developed as a direct result of philosophical inquiry. See the collective works of Francis Bacon and René Descartes, two of its most notable founding fathers.
I'll try to answer in 2 ways, since you gave your own "Scientific Method in a nutshell" definition:
1 - As you put it, the fact that causation is central to the objective of SM ends up putting Science and Philosophy nearer than what seems to be your vision given the title question, since "causation" and "why" are connected ideas.
Seeing things this way, I'd say that yes: a philosophical explanation about something testable will get the benefits of SM, although maybe it's trickier to rule out concurrent philosophical theories by experimentation.
About the why and how thing, I don't think it's a good way to understand the difference between Philosophy and Science, but assuming it is, I suppose that "how" is more tightly linked with the data. "Why", in its turn, admits different ways of explanation. This would, in fact, mean that the trustworthiness of SM would be weaker when applied to Philosophy.
2 - The wikipedia¹ takes the Oxford English Dictionary definition of SM to be:
"a method or procedure that has characterized natural science since the 17th century, consisting in systematic observation, measurement, and experiment, and the formulation, testing, and modification of hypotheses."
If we give up the How × Why distinction, I'd say again that yes: we can apply SM to Philosophy where all (or at least most) of necessary parts are present. If philosophy rarely applies full SM, it's because the studied subjects are more speculative and difficult to "square" in the SM framework.
About the trustworthiness, I don't see any problem "by definition", although if we don't have all the component parts of the method, we'll not get it to exert its full power.
One should have in mind that this kind of question is always puzzled with the problems of defining what is philosophy. Maybe some of the appealing character of this question comes from the fact that some people who try to define philosophy take as one of the basic characteristics the non-experimental factor, so this opposition (with which I don't agree) emerges.