A quick qualification: Laruelle's decades-spanning project is going to be something very difficult to convey accurately in a few paragraphs, especially when trying to field the supplemental questions you pose here. It might make sense to try to approach all this a bit more straightforwardly, slowly, incrementally -- if a careful, pedagogical explanation of the material is actually what you're after :)
Another qualification: while I am not really an expert on Laruelle, I have attempted to read it in good faith, and have plugged through a lot of his work that's available in English. Finally, I do find myself quite sympathetic to his project.
More or less the basis of my own interest in Laruelle, and at the very least a key reference, is Deleuze and Guattari who at the end of What is Philosophy? talk about nonart, nonscience, and nonphilosophy as "outsides" of art, science, philosophy. These negations form part of a shadow which renders art/science/philosophy indiscernible; the shadow of a future people, a people to come, capable of advancing beyond the arbitrary/neurotic/anthropoid divisions between these creational practices (towards a reunified/indivisible art-science-philosophy, something like what Nietzsche talks about with his joyful wisdom which would reunite art and science once they have sufficiently "matured".) Philosophy needs a nonphilosophy to comprehend it, D+G say. It is definitely a nod to Laruelle and they mention him as one of the most interesting working contemporary French philosophers in a footnote (note this was some 20 years ago.)
Okay. So let's start thinking about decisions, or as you emphasize distinctions or "dialectical splittings". This may seem odd, but Laruelle's point that all philosophies are structured around a prior decision is not an "axiom" for him; rather it is the axiomatic character of philosophy that is here captured in a kind of non-axiomatic way. The decisional structure of philosophy demands not an analysis based on assumptions; but as he says a kind of "dualysis"...
Laruelle identifies something like a Godel-effect operating over the informal-abstract conceptual phase-spaces through which philosophy traces its cognitive trajectories, constructs its planes of reference, effracts new conceptual schema, etc. There is a hole in every philosophical plane of thetic organization. Concepts or system of concepts which populate these planes, and the discourses disclosing them, cannot avoid these vortices. Philosophy inevitably relies on obscure distinctions (confused decisions) that cannot be grasped by that philosophy in itself, or rather that require new concepts ad infinitum. I am tempted here to talk about the fractality of sense or thought, the fragmentation of philosophical "dimensionality". But again this is really endemic to an image of thought called "philosophy" (Aristotle says Zeno invented the dialectic with his paradoxes of motion.)
One key to my mind is the "non-Marxism" of Laruelle; as Deleuze says, for most of history, philosophy amounts to an image of thought that enforces stupidity, that actuallys stops people from thinking. Laruelle says something to the effect that future generations may not show the same gratitude towards philosophy -- and its repressive, neurotic, stunting effect on human thought and ultimately life on the planet -- that we are still hypnotized into showing today. Nonphilosophy is a kind of generalization of Marxism, at its humanist core a project about improving the condition of humanity in the world. For Laruelle, radical immanence is "humanity in person".
So Laruelle posits the "transcendental" equivalence of all philosophical positions: a kind of "suspension" of certain philosophical axioms in order to open up a space for thought, experimentation, learning, creation, etc., outside the (regressive, exploitative, philosophical) image of thinking, knowing, creating; making use of philosophical materials in new ways...
Hopefully this represents some starting points and helps give a sense of the spirit of the work. Apologies for the obscurity/sparseness of this presentation (it is going to be difficult to summarize his project briefly, and it may be more constructive to approach the subject a bit more iteratively; please consider asking further questions here!) Laruelle's work is definitely difficult, but can also be incredibly rewarding; I would definitely recommend looking at Philosophy and Non-Philosophy and Principles of Non-Philosophy, both of which should now be available in English translation.