Do contemporary logicians generally claim, as Wikipedia does, that classical logic can be simply reduced to the 5 logical principles below? Or is it more complex than that and are there principles not included?
I take your question to be, are the listed features sufficient to precisely and completely distinguish classical logic from all non-classical logics? There are certainly lots of other things one would need to specify in order for classical logic to qualify as a logic at all. And there are many other distinguishing features of classical logic not listed that follow from the ones that are listed.
The problem with giving a definitive answer to this question is that it is a matter of disagreement over exactly what constitutes a classical logic. Some use the term in a limited fashion to include only elementary first-order predicate logic. Others use it in a more extended way to include any logic that has the property that all theorems of elementary classical logic are theorems of that logic. These come apart in a number of ways, some of which are:
The standard semantics of classical logic is bivalent, which is to say it is the logic of pairs of opposite values, usually true/false, such that every proposition has one and only one value. But some accounts allow that multi-valued logics can be classical, as long as these logics are strictly extensions of bivalent classical logic and do not conflict with it.
Modal logics are commonly treated as non-classical in modern textbooks, but usually they extend classical logic and so would count as classical logics in the extended sense.
The identity relation is often treated as part of the logic, rather than as a predicate that can be interpreted as part of a theory. One speaks of first-order predicate logic with or without identity.
There are standard rules for handling variables, quantifiers and domains in classical logic. These can be extended in various ways, e.g. with plural quantification, branching quantifiers, typed variables, etc. Some of these extensions might be considered classical, but others, such as the ability in free logics to quantify over non-existent objects, are usually considered non-classical.
First order logic can be extended to second order and higher orders. Some logicians (famously Quine) do not consider these to qualify as logic at all.
The listed features do not stipulate bivalence, do not exclude non-truth-functional modal operators, do not specify any identity relationships, do not place any restrictions on quantification, and do not specify first-order, so they are obviously intended to characterise classical logic in an extended sense.
The answer to your question is at the start of the article you cited:
Each logical system in this class shares [these 5] characteristic properties... While not entailed by the preceding conditions [emphasis mine], contemporary discussions of classical logic normally only include propositional and first-order logics. Classical logic generally entails the majority of the Western tradition of philosophy and logic since the pre-Socratics, but particularly starting with Aristotle.
"While not entailed by the preceding conditions" means that 'there is more than these 5 conditions to describe classical logical study'. Five sentences would hardly be enough to describe classical logic.
What is classical logic? According to the Encyclopedia of Philosophy and it's entry 'Logic, Non-Classical':
[C]lassical logic [is] the theory of validity concerning truth functions and first-order quantifiers likely to be found in introductory textbooks of formal logic at the end of the twentieth century.
One way to define what classic logic is is to define it as what non-classic logic is not. (See PhilSE: In how many and which ways can a logic be non-classical? Are there systems for organizing them? for a list of systems and their characteristics.) This is why the principles are featured prominently in this article because what non-classic logic is about is deviation from some of these principles. For instance, sequent calculus and its extensions such as display calculi, nested sequent systems, and labeled sequent systems are metalogical and go beyond classic-logic as metalogical theory. But it's much more informative to describe some of the features of classic logic and some of their historical origin.
Is there more than the principles? Well as the WP article concedes, yes. In fact, of the three generally acknowledged Laws of Thought, one is not mentioned in the list of principles above: the Law of Identity. So, immediately, the moment discussion starts about classic logic, the Law of Identity is an important topic; the famous philosopher of language, John Searle, for instance, challenged the Law of Identity as a law arguing it is only a convention in his paper Proper Names. (See PhilSE: On Searle's _Proper Names_ (1958) for an explanation how he invokes the conventions of cryptography to challenge it.)
From Volume 5 of the Encyclopedia of Philosophy, the entry "Logic, Traditional" begins with this:
In logic, as in other fields, whenever there have been spectacular changes and advances, the logic that was current in the preceding period has been described as "old" or "traditional"...In every case, the logic termed "old" or "traditional" has been essentially Aristotelian, but with a certain concentration on the central portion of the Aristotelian corpus, the theory of categorical syllogism... especially of the sixteenth to the nineteenth century.
In fact, WP's entry on the History of Logic has a subsection on traditional logic:
Other works in the textbook tradition include Isaac Watts's Logick: Or, the Right Use of Reason (1725), Richard Whately's Logic (1826), and John Stuart Mill's A System of Logic (1843). Although the latter was one of the last great works in the tradition, Mill's view that the foundations of logic lie in introspection influenced the view that logic is best understood as a branch of psychology, a view which dominated the next fifty years of its development, especially in Germany.
So, besides missing the Law of Identity, here are some additional topics: categorical syllogism, the logic of propositions, equipollence, and Euler's diagrams, as well as the history of classical logic more generally to understand the context of their development and relations. In fact, two of the most important works are from three heavyweights of the analytical tradition. According to that WP article:
A more technical overview can be found at SEP: Classical Logic.
No. The Principle of explosion is absurd, i.e. illogical.
And then the fact that Monotonicity of entailment is false follows from the absurdity of the Principle of Explosion.
The so-called "classical Logic" part of mathematical logic is in contradiction to Aristotle's logic, which is the only logic which can be properly called "classical".