Is there any exception or rebuttal to the law of noncontradiction? Do we know for certain that this law is universally true, or are there some situations where the law simply doesn't apply or is false? I take this law for granted and to be absolutely true, but I am wondering if there are differing views on this.
As common sense as you intuited, law of noncontradiction (LNC) is considered to be necessarily true universally (in all possible worlds) from which analytic statements follow from by most philosophers such as Aristotle who asserted the most certain of all basic logic principles is that contradictory propositions are not true simultaneously. However, this is not immune to be attacked from skeptics from modern logical pluralism holding the view that there is more than one correct logic according to SEP reference here:
Different logics disagree about which argument forms are valid. For example, logics like Classical and Strong Kleene logic tell us that that ex falso quodlibet (principle of explosion), the argument form below, is valid: A,¬A ⊢ B. However Relevant logics and other Paraconsistent logics say that this argument form is not valid. It’s natural to think that they can’t all be right. If ex falso quodlibet is valid, then the Relevant and Paraconsistent logics are not correct theories of validity, or as we might say, they are not correct logics. Alternatively, if ex falso quodlibet is not valid, then Classical logic and Strong Kleene logic are not correct. Logical pluralism takes many forms, but the most philosophically interesting and controversial forms of the view hold that more than one logic can be correct, that is: logics L1 and L2 can disagree about which arguments are valid, and both can be getting things right.
So in summary within a single logic system (be it classical, Kleene, intuitionistic, relevant, or other paraconsistent logic), LNC has been accepted to be true so far, but the same argument proposition can be viewed as both true and false under different logic system one chooses to adopt, such as above principle of explosion argument is true under classic propositional logic but is false under modern relevant logic proposed by Orlov since the contradictory antecedents may not have any relevance to an arbitrary consequent thus explosion is avoided.
Having said that, another similar basic law of thought, that is, law of excluded middle (LEM) is heavily challenged within mathematics by the modern constructivism school led by Brouwer against Hilbert's formalism according to reference here:
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not include the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic. Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for Brouwer's programme of intuitionism. From a proof-theoretic perspective, Heyting’s calculus is a restriction of classical logic in which the law of excluded middle and double negation elimination have been removed.
The rule is definitional, of a system that does not include framings of identity that permit this. Rather than say, discovered by observing the world, or somehow directly known by introspection.
See for instance Buddhist four-valued logic which is much better at paradoxes. See how Nagarjuna uses it. And more generally Buddhist 'anatta' and deconstruction of fixed identities through contemplating dependent origination, as illustrated in the Indra's net metaphor.
The ship of Theseus, and teleporter paradoxes, are examples of challenging conventional or intuitive notions of identity in Western philosophy.
No, there is no exceptions. All purported "rebuttals" of the law of contradiction are based on the same fallacy, the fallacy of equivocation. Deductive logic is defined as based on the law of contradiction. If you somehow decide that this law doesn't apply, then you are not being logical, and the words you will use in this context will no longer mean what they mean for people who are reasoning logically. Cheap trick.
We are free to choose not to be logical, and even to pretend that we are being logical precisely at the moment we are not, but it is easy to verify that either the person is mad or inconsistent, inconsistent in the sense that they will at some point pretend to be able to deny the law of contradiction while abiding by it the rest of the time, in particular whenever the question turns on their legal rights and they property.
The law itself is based on the semantics of negation and the semantics of negation is an intrinsic part of the way the human mind works. We have to distinguish in this respect how someone's mind work and what this person chooses to say. We can all say we believe things that in fact we don't. We can all tell lies.
The equivocation involved in denying the law of contradiction is the equivocation about negation. Saying that A ∧ ¬A is true (for some A) is to equivocate on the notion of negation. Negation means something in the context of the law of contraction and something else when you deny the law of contradiction.
This is also equivocating on the word "logic". The word "logic" means something when talking about the logic of human deductive reasoning, and it means something else when you are not talking about the logic of human deductive reasoning.