My question is does this philosophy have something specific to say about the initial conditions of the universe?
It certainly does. The Father, First or Source would be self-generated and constituted as or by its own Knowledge. This is what the mystics mean when they say Knowing is fundamental. In the Gnostic explanation the First has three powers that are intrinsic. These powers are projected to form the Divine Intellect, which is constituted by the contemplation of its source. All else would follow.
I think you'll find that this explanation does away with the distinction between initial conditions and laws of physics in the way you suggest. By reduction the laws become expressions of what is, not separate from their source.
I have not seen a detailed explanation in the advaita literature but the Nag Hammadi Library has the text of the Zostrianos which gives an astonishingly detailed description. It explains that the projection of the Intellect and subsequent symmetry-breaking that leads to the emergence of form from formlessness is not the creation of more 'things' but an error of perception and a forgetfulness of our Origin.
There would not be two things. The First would encompass All just as Plotinus describes. The initial condition of the world would always be its underlying state such that nothing really exists or really happens and this would be the reason why we are able to know our Origin. But it can only be known as what it is so can only be verified or known by 'becoming'.
The laws of physics would be intellectual laws and this would be why they make sense to our intellect and are mathematically and logically sound. These laws, Lao Tsu tells us, would be as they are 'Tao being what it is'. Thus from the initial condition follows all the rest. For the mystic the world is Euclidean and so is its correct explanation.
The initial condition for physics is explained also by George Spencer Brown in his Laws of Form where he describes the emergence of form from formlessness. There are many other descriptions but the description in the Zostrianos is the best I've ever seen.
Have you checked out the advaita teacher Wei Wu Wei? He became a friend of Spencer Brown. I seem to remember he has some things to say about this and is always worth reading anyway. A follower of his is Ramesh Balsekar whose book The Ultimate Understanding you may also find useful.
They are, of course, describing the indescribable so we cannot expect to make much sense of all this without actually getting our hands dirty and indulging in the practices of advaita and self-enquiry. It would not be necessary to attain complete enlightenment to make some sense of it, but we would probably need to get our feet wet.
EDIT: Just noticed I didn't talk about 'all-pervading'. The initial condition would be all-pervading because nothing else would be truly or independently real. Space-time would be conceptual. In my speculations this would explain the weirdness of QM and in particular 'non-locality', which would be better called 'locality' because there would not be two places or two times.