Is Logic Empirical?

Not exactly. Of course, there is a broad reading of "empirical", which includes anything somehow extracted from experience, upon which the answer is trivially yes. But on this reading God is also empirical because some people experience communicating with him. On the more conventional meaning of "empirical", the opposite of empirical is not necessarily innate, absolute, eternal or fundamental. In particular, non-empirical can be fallible. A law is empirical if it can be "derived" from observations by induction/generalization, like Kepler's laws from observing the heavens. The laws of logic are not of course unrelated to experience, or "absolute", but they can not be so derived. See also Is geometry mathematical or empirical?

At the dawn of modern psychology, in the 19th century, the opposite point of view was advocated by many of its founders, and even by some philosophers like Mill. It came to be known as psychologism about logic, and some of the arguments were elaborations of yours. It died out after Frege and Husserl showed that it leads to inconsistencies. First of all, the certainty of logical laws, while not absolute, far exceeds that of any psychological laws. So the former can't possibly justify the latter. Second, if logic is "subjective" it is a miracle how we manage to communicate successfully while using it. Here is more from Husserl's Prolegomena:

"Extreme empiricism, therefore, since it only basically puts full trust in singular judgements of experience... eo ipso abandons all hope of rationally justifying mediate knowledge. It will not acknowledge as immediate insights, and as given truths, the ultimate principles on which the justification of mediate knowledge depends; it thinks it can do better by deriving them from experience and induction... If, however, all proof rests on principles governing its procedure, and if its final justification involves an appeal to such principles, then we should either be involved in a circle or in an infinite regress..."

In other words, "deriving" logic from empirical experience involves employing the logic itself in the derivation. After the downfall of psychologism logical positivists offered an alternative proposal, that logic is adopted by convention. It is interesting that it fails for essentially the same reason, as Quine pointed out in Truth by Convention:"In a word, the difficulty is that if logic is to proceed mediately from conventions, logic is needed for inferring logic from the conventions".

So if logic is neither empirical nor conventional, what is it then? First, we need to distinguish, after Peirce and scholastics, two different logics: "logica docens" (doctoral logic), and "logica utens" (logic in possession), of instinctive reasoning. Like Kant's "a priori" Euclidean geometry, the latter may very well be biologically innate, or imprinted in early childhood, while the former, like axiomatic geometry, is chosen and followed systematically. But even the logica docens is neither empirical, nor subjective, nor conventional. The word is "theoretical" or "constitutive". Like theoretical entities (quarks) and constitutive laws (Newton's second) logic is not observed, measured, or inferred from experiments, it has to be adopted prior to experiments to make sense of what is observed, measured, or inferred.

But while it is a priori, it is not eternal or infallible. "Falsifying" logic empirically is not exactly meaningful, but we may still choose to abandon it if it is deemed counterproductive overall using holistic (i.e. extra-empirical) criteria of theory selection, and adopt an alternative. So far, this did not happen with quantum logic, after a burst of interest in 1970s it faded into a niche activity. But there is a recent proposal, which would adopt it as the logic of quantum gravity substrate, out of which the spacetime emerges, see Raptis' Presheaves, Sheaves and their Topoi in Quantum Gravity and Quantum Logic.

There are quote a few questions in your question, so I'm just going to amplify on one.

First, some-one, I forget whom, quipped that mathematicians take something, turn it into their own language and then it's something completely different. This, though a quip, has a kernel of truth to it.

Mathematical logic is different from logic per SE; merely by formalising it one is forced to make choices, and then later one can argue about these choices; for example, should one formalise the principle of 'explosion'? That one inconsistency renders all sentences inconsistent?

Logic has, historically several sources, and one of this is language, where we state propositions; here if I state an inconsistency, we are hardly going to say well, everything that I have said and everything that I will say will be inconsistent. Instead, we suppose I made an error - either deliberately, or unknowingly; what we see here is that logic is understanding how to reason correctly.

Another sense, as pointed out by Heidegger associates logic with ontology; this was at first by Plato, and then much later and in a different form by Hegel. It's this echo of ontology in logic that probably leant its support for the principle of explosion in formal logic.

To return to the main point, once logic is formalised we can look for resemblences in a merely formal manner - and this is a mode of thinking that wasn't open before; further, we can drop or add formal laws as we see fit, and this again a new possibility.

This is how the old quantum logic - by Birkhoff and Von Neumann was discovered; the new quantum logic isolates certain features of QM and thinks them through categorical logic; interpreted, they are a logic of processes - and here phenomena as no cloning, teleportation, or entanglement become more perspicuous.

The point is that formal logic, merely by its formality, requires interpretation for it to make any sense; and this might be a tenuous connection to the classical concepts of truth and falsity - which might help us make a renewed acquaintence with these somewhat jaded and out-moded concepts - or just as possible, be more offhand with them.

Excuse me, but first I should correct this statement:

In Buddhist Tradition Dīgha Nikāya provides an example. As the Buddha explains in the Brahmajāla Sutta, there are four alternatives: (1) The world is finite, this is one case. (2) The world is not finite, this is another case. (3) The world is both finite and infinite, this is the third case. (4) The world is neither finite nor infinite, this is the fourth case. (5) There are no other cases.

Buddha in Brahmajāla Sutta didn't say there there is four alternatives about World's finiteness, he says that some recluses and brahmins have have such and such opinions. Difference is, that Buddha describes other's opinions, and not stating his opinion. In this tetralemma position is nothing particularly Buddhist, this is just Indian logic at the time (see Catuskoti).

Why it's fourfold and not twofold (as is usual in A and ~A). Because, it is logic of natural language and not formal logic. In formal logic we have rule of identity to be also represented in language, so we can state that what is written as A is always meant as A (and vice versa), so there is only ~A possible as alternative (in formal writings). In natural language, we don't have such important convention, so it is possible to write that thing is A (in some sense), and in same time not A (in other sense), where sense of such writing should be derived implicitly. Non-formal writing would look contradictory if we miss context (i.e. if we misinterpret it). As a consequence, there is four alternatives instead of two to cover all possible 'syntax'.

For Example Let us take the quantum logic , in quantum logic An electron can be both at Position A and B at the same time. Classical Logic does not permit so.

Why it does not permit so? Just assume electron can have both positions and voila, it's permitted. Logic does not say that if electron have position A it can not have position B (it does not know anything about electrons). It's you saying that. If we put not valid propositions into logical inference it will produce not valid conclusion. It is not guilt of logic, because logic is method for avoiding mistakes in inference. Logic warns you to take correct propositions before inferring, so, if you put object which can have two positions and then assume it could have only one, this is your fault, not of classical logic.

If you can not construct mental object for which A=A is true, you can not put it into logical inference, because you breaking identity rule. Logic can not infer anything you would wish, it can only infer some valid things if you obey it's rules.

Logic also does not vary. Why there is multiple Logics then? Because this is multiple formalizations of logic and not (variance) of logic itself.

And finally, if A is bigger than B and B is bigger than C, is A bigger than C? When you know this is it empirical to you?

I'd say that logical knowledge is empirical only in the sense that while conjectures might be shown to be true or false by knowledge of the world around us, this can happen in an empirical way only by holding up specific examples. However, this only has so much explanatory power in logic and mathematics, as some generalities fail to be proven or disproven by specific examples - take, for example, Fermat's Last Theorem. This type of knowledge - knowledge of generalities, is a priori, if such a distinction is allowed.

The real numbers are a purely theoretical construct. While they were motivated by "real world" concerns, they exist completely independently of it. There is no* empiricism in their study.

However, they are a tool that can be applied to describe features of physical theories, and how to do so could be said to be empirical.

Similarly, lattices of logical propositions are purely theoretical constructs, but they can be applied to describe features of physical theories.

That's all that's going on here. It's not very different from doing geometry via vector calculus rather than by manipulating coordinates.

*: Not strictly true; empirical methods are a part of the mathematician's toolbox. (e.g. gathering numerical evidence to formulate a conjecture)