Contrary to the basic notions of Classical thought, what logic you need really does depend on what you are doing.
Classical logic uses the material conditional, your first option, because it is relatively safe to do so. It is part of a simplified model, and in that model, this way of accounting truth does not lead to contradictions we do not like. However, it does not correctly model deduction, and it is particularly unrealistic in modeling situations where deductions are made with limited information.
If you use the axiom A => (B -> A) directly in real life, you are making no sense. The true fact that most elephants are not yellow is not a consequence, in any natural way, of the fact that some monkeys like sugar. But the former implies the latter in Classical logic, or in a computer circuit or a decision in Newtonian physics. It is an axiom of most logical systems.
It summarizes the idea that we don't really know the exact deductive path from one true fact to another true fact, but we don't feel we need to. We trust fields like math and basic science, and even large chunks of philosophy, to a certain degree to be timeless and complete in a way that makes all true things equally true.
In those contexts, which make up most contexts, the answer to (1) is that it is true except in bizarre circumstances. There are ways, see Curry's paradox, in which we can get weird results using these axioms. But we would notice those right off. We are safe using this logic in deductions about everyday things.
In contexts like law, where judges do not know what decisions other judges are making at the same time, or have made in distant jurisdictions, this principle does not hold. There can ultimately be a contradiction between their decisions in the future. You cannot simply depend on the material conditional to tell you whether something is true.
Otherwise, you could not lose now and win on appeal, unless the logic of the first case were found provably incorrect. But we see it happen -- decisions change, for good reason, that were, at the time they were made, right. There is just limited information that requires real arguments, using a more 'real' notion of implication, to flow together in a more literal way. So for instances like that, we need ways of reasoning that are less abstract than the ones in ordinary mathematics.
The same problem occurs in fields of physics like quantum dynamics, and in reasoning about other people psychologically, where things can only be found consistent to a limited degree. Sometimes the simplified rules just don't cut it.
So the answer to (2) is that you can really follow the threads of an argument as closely as you need to, when you need to. We spend the time to do it in these domains, and we are confident of the results. But if there were a short, constructive answer to (2), we would never have invented Classical logic to begin with.