These are interesting questions and quite difficult to answer. There are many different logics, and as you say, one's view of logic connects with how one thinks about the world.
When most people think about the "real world" they think of it as existing independently of ourselves, and of facts about it being true whether or not they are known, or even knowable. Other domains, such as mathematics, or ethics, or aesthetics, might be viewed differently. In the history of philosophy, all kinds of debates have arisen about whether certain things are 'real'. Depending on the context, realism can be contrasted with idealism, nominalism, constructivism, phenomenalism, intuitionism, instrumentalism, etc. Michael Dummett introduced the neutral term "anti-realism" to refer to all these. He then made an important observation that connects realism with logic. Our willingness to be 'realist' about a particular domain corresponds to our willingness to accept that a statement about it is either true or false, irrespective of whether it is verifiable, or indeed whether any evidence exists about it. For example, consider the sentence, "it was raining on Manhattan island at noon on 1st May, 10,000 BCE." We have no evidence as to whether this is true, and it seems likely that we never will. But most people are inclined to say that it either rained or it didn't: this is just a fact about the world that is independent of whatever evidence we have.
In other domains, such as mathematics, this is less compelling. Some mathematicians are realists (Gödel was), but it seems plausible to say that mathematics is about what can be proved; there is little reason to think that all mathematical sentences must be either true or false, even when no proof exists.
This suggests that classical logic is the natural logic of realism, while intuitionistic logic, which lacks the law of the excluded middle, is naturally an anti-realist logic. There are logicians who maintain that there is a single 'correct' logic. Michael Dummett defended intuitionistic logic in this way, Stephen Read argues for relevance logic, and Graham Priest for paraconsistent logic. But it is also possible to understand these logics as having a different natural semantics. Classical logic is about truth, and classical validity preserves truth; intuitionistic logic is about verifiability; relevance logic has the semantics of information passing; paraconsistent logic has different flavours, but at least one can be thought of as having the semantics of falsifiability. S5 is about logical necessity; S4 has the natural semantics of a weaker form of necessity that is consistent with branching possibilities.
Understood like this, one could use several different logics, without having to accept one as being the only correct one. It still leaves open the question as to whether there is a natural logic of the metalanguage. What logic am I assuming right now when I talk about logics?