You are grappling with the idea of logical and truth pluralism. This is a question that philosophy has been trying to come to terms with for a century and a half, with much of that time spent with loads of philosophers denying the issue, which has slowed the development of an answer.
The consensus view in the mid 1800s is there is one math, and one logic, and they are both true by necessity. Kant articulated this view.
However, non-Euclidean geometry was developed and shown to be self-consistent, and as this was Kant's go-to example of necessity -- the necessity view of math was refuted. Necessarians tried to hold on to "at least the world is Euclidean", which would make math true a posteriori, rather than a priori (this is a peculiar position for math, but straws were being grasped). But then Einstein blew that weak fall-back out of the water too. The consensus among mathematicians is that math is real (abstract object plationism), but that there are LOTS of maths, and the choice of what particular math to apply is a formalism, no specific form is "necessary". This is mathematical pluralism. The preference for a particular math is PRAGMATIC, not based on logic.
Physicists embraced this approach to math, and as logic and math are basically the same category of system, one should expect pluralism to apply to logic as well. And a century ago, Quantum Mechanics was developed with that as an assumption. Quantum math does not follow classical logic. The double slit experiment, the Heisenberg uncertainty principle, entanglement, Bells Inequality -- these are all based on non-classical logic.
As you note, this brings up the question of ultimate causes, and the validity of "nothing comes from nothing". And sure enough, physicists, who embraced plural logic pretty early, propose "nothing from nothing" events. Hoyle's "Steady State" universe existed forever, but the MATTER in it -- came from nothing. He proposed that there was a small rate of spontaneous proton formation in empty space. And Hoyle is hardly alone. Guth's proposal of "inflation", in which mass and space just form spontaneously due to an "inflation field" is assumed in pretty much all subsequent cosmology. Susskind's Cosmic Landscape expands this to the spontaneous formation of an infinite multiverse, not just our universe. And while Guth and Susskind did not give up on causation (they start with a seed "field") Hawking did -- in A Brief history of Time he proposes that the universe is "a closed shape in spacetime", IE nothing before (no cause) and nothing after (no consequence).
Logicians have been slow to accept this thinking. But over the last several decades, logical pluralism has become the consensus. A useful discussion is here: https://arxiv.org/abs/0705.1367 (the PDF download is free).
As you note -- this brings TRUTH into question. Many people, including philosophers, resist the radical consequences of pluralistic logic, because they don't see any other way to get to "truth". But there is an alternative -- it is to approach truth pragmatically. Formal logic is "very useful" therefore one should generally accept what it demonstrates, as tentatively true, unless one has significant rationale or justifications to think otherwise. Truth is uncertain, and the means to find it is uncertain, but we have a lot of tools which are highly useful, and generally bring us to "good enough" approximations of truth.
So -- your top question -- the answer is "no". We cannot know that non-contradiction is true a priori, and based on the intrinsically pluralistic nature of logic, we can actually know it is NOT always true, and know this a priori.