Logic, mathematics, science, musicology, and most systems of government and commerce are all examples of attempts to create rigor and consistency in our thinking to address the flaws of our built-in logic inherited from our mammalian ancestors.
We are born using a form of Bayesian logic or abduction described in the Theory of Sufficient Reasoning. Immortalized in "Occum's Razor" we gravitate to the simplest explanation, drawn from no more than our personal history of experiences. We judge our experiences and form a collection beliefs in a world view that acts like a filter or paradigm through which we judge future events. Each belief carries with it a credence value or disposition regarding how strongly the belief is held. When an event is witnessed displaying evidence in favor of or against the belief, the rational person will then adjust the credences affected accordingly based on their judgement as conditioned by the prior credence values. https://en.wikipedia.org/wiki/Abductive_reasoning
The strength of abductive logic is that we can make quick decisions without perfect or complete knowledge of the facts, a very valuable tool for survival. This survival skill, however, comes with obvious flaws. Quick decisions without sufficient data results in tragic errors. Also, chronic, traumatic or systematic conditioning can set credences so high that one may not be able to recognize new evidence let alone judge it. Lastly, a credence change cannot be made without maintaining consistency in one's world view. If changing one credence causes unacceptable changes in other, the original belief change may be set aside, thereby weakening the authenticity of one's world view.
So, the logic systems found in scientific, mathematical, and philosophical systems do impose rules of engagement to ensure integrity and reliability and to avoid or mitigate the flaws in abduction. Ironically, when scientific or mathematical discoveries are offered, abduction takes full charge in judging its acceptability. Resistance to change that threatens the system underpinning will be vociferous, even though the logic and evidence are compelling.
Logic systems are like scientific theories. If they consistently provide the right answers in their assigned domains, we accept and use them. But, what happens when a contradiction appears?
Your final concern, whether our choice of logic affects our math and other pursuits is a resounding affirmative. Aristotle's logic based on the premise that a statement must be either true or false created the logical foundation for the industrial revolution and remain embedded in every institution even though it is patently and inhumanly wrong. (Any Charles Dickens novel)
Ultimately, you may master these systems and still have doubt about their credibility because your world view objects regardless of the evidence. On the other hand, you may find, as you learn more, that your inherit logic system can serve you well with help from other logical points of view. I suggest that you suspend judgement for a while, and allow your natural curiosity to seek out the answers to your concerns with knowledge.