Since /u/Damasus222 already covered your questions about Sokrates and about the state of Greek culture after the war, I can focus on your initial questions about the war itself. I just want to add the point that there's a widespread modern perception that the heyday of Greek culture ended in 404 BC, which is wrapped up with modern worship of Thucydides and of Perikles and Periklean Athens. Even academic scholars frequently argue that the Greeks reached the height of their splendor and sophistication under Perikles and it was all downhill from there. Never mind the fact that the great majority of the surviving literary sources were written after the Peloponnesian War, that Athenian democracy was more durable and stable in the fourth century BC than the fifth, that evidence of all kinds suggests Greece in general was more populous and prosperous in the fourth century BC than the fifth, and so on. Admittedly some of the idea of the "decline of the polis" after the war comes from fourth-century sources like Isokrates and Demosthenes, but these were orators with axes to grind, and it says something that their even survives; with the exception of Andokides we have no such texts at all from the fifth century.

Now, the Peloponnesian War. You asked:

How bad was the situation in Athens in order for the Spartans to react as they did? Simply from a commonsensical point of view, starting a brutal, bloody war with a different tribe from the same land as yours sounds like a stupid idea in any case and I don't think it has happened at any time before in history, so the incentives for it must have been massive. All the textbooks I have read seem to downplay it. Maybe I am not thinking Greek enough, but "one city which suffered the most in the war with Persia taking taxes from other cities in order to compensate" doesn't sound as a justified enough reason to start a civilization-ending war.

There are a lot of flawed assumptions here that are probably the cause of your confusion. I'm going to go through the paragraph to unpack them in the hope that it will explain why the Peloponnesian War broke out.

First, "a different tribe from the same land". While there was a collective Greek identity at the end of the fifth century BC, the Greeks did not regard the Peloponnesian War in any sense as a civil war. The Greeks lived in fully autonomous communities that acted as independent state actors on the geopolitical stage. Indeed, the war broke out in part because some of Athens' subject allies found it intolerable that the rise of the Athenian Empire meant that they could no longer choose their own friends and enemies. Wars between Greek states were normal, accepted and common; it was unusual for Greek states to be stripped of the freedom to declare war on each other whenever they wished.

In fact, the Peloponnesian War was not a sudden outbreak of catastrophic war in a formerly peaceful environment. What officially happened in 431 BC was that Sparta rescinded the 30-year truce it had agreed with Athens in 445 BC. That is, Athens and Sparta had been at war in the recent past; they could not agree to a lasting peace, but only agreed to suspend hostilities for 30 years; in the end, they couldn't even get through that period, but went to war again after just 14 years. So much for the idea that there had been no such war before in history. (If you would like to know more about the so-called First Peloponnesian War of 460-445 BC, I recommend you read my answer in this thread.)

But the causes of the war, as summed up by Thucydides, also show that there were other conflicts going on. He sums up four "grievances" that provoked the Spartans to war (one of which he downplays to protect the reputation of Perikles):

1. In 433 BC, the Corinthians went to war against their colonists on Kerkyra, but the Athenians made an alliance with Kerkyra and sent ships to help them. The naval battle at Sybota ended in defeat for Corinth, which complained to its hegemon Sparta that Athens was meddling in its affairs.

2. In 432 BC, the Corinthian colony of Potidaia (part of the Athenian empire) revolted from Athens. Corinth sent "volunteers" to defend the city against Athenian reprisals, but they lost the battle outside the city and were placed under siege. Corinth again complained to Sparta that Athens was infringing on its interests.

3. In 458 BC the Athenians invaded their ancestral enemy Aegina, incorporated it in their empire and subjected it to heavy tribute. Aegina complained to Sparta that the Athenian Empire began as a voluntary alliance but had now become a tool of coercion and oppression.

4. In 447/6 BC the city of Megara had betrayed Athens and taken the Spartan side in the First Peloponnesian War. In response, in the later 430s BC, Perikles persuaded the Athenians to pass the Megarian Decree, which barred the Megarians from all harbours of the Athenian Empire and forbade its allies to trade with them. Megara complained to Sparta that it was being squeezed and that its hegemon ought to stand up for a vulnerable subject ally. Thucydides barely mentions the Decree but other sources suggest this was the main cause of the war.

These were the concrete reasons why the Peloponnesian War broke out. As you can see, none of them have anything directly to do with Sparta. But all of them have to do with Sparta's reputation as the leader of all the Greek states. Sparta's allies were complaining to them and urging them to act against Athens; with every day that passed, the Spartans looked weaker and less able to defend their supreme status among the Greeks. Sparta was still in control of a large network of subject allies known as the Peloponnesian League, but if the states of this League ever became convinced that Sparta would not protect them, Spartan power would swiftly crumble. At some point they knew they would have to act.

This is why Thucydides said that the "truest cause" of the war was not any of the specific grievances with which the Corinthians, Aeginetans and Megarians lobbied for war. It wasn't even the nature of the Athenian Empire itself, and its previous violent repression of uprisings in Naxos, Thasos, Miletos, Erythrai and Samos. In his view, the "truest cause" (that is, not the one true cause but the one that was most true of all) was that Athenian power was growing and the Spartans were afraid. He does not spell out what it was exactly that the Spartans were afraid of, and certainly Athens was doing nothing to hurt Sparta itself. But it seems most likely that they feared the gradual diminishing of their status as their allies were pushed back by Athens and came to realise that the Spartans would not stand up to defend them. They may also have feared that if they failed to act soon, they would no longer be able to gather enough military strength to challenge Athenian supremacy. In 432 BC, a tipping point was reached, and the Spartans ordered the Peloponnesian League to prepare for war.

I'll start with the second half of your question, about Socrates, culture, and Sparta. There are others around here much more qualified than myself to treat the first portion about the war proper and its causes.

First, to clarify, Greek philosophy and poetry most definitely do not stop at the end of the Peloponnesian War. Cynicism, Epicureanism, Stoicism, and other philosophical schools developed in the fourth and third centuries BC. Not only are there post-fifth century poets, but there are entire genres of poetry (e.g. New Comedy, bucolic poems, epigrams) that are either invented or began to flourish in the two centuries following the war. So poetry, philosophy, and "high" culture in general all continue to grow and change after the war, as they will through the Hellenistic period and under Roman rule.

Now, on to Socrates, who requires a bit of backstory. After Athens surrendered to Sparta, the Spartans replaced the Athenian democracy with an oligarchic regime headed by thirty wealthy Athenians. The Thirty ruled in a draconian manner, showing little compunction about executing those they perceived to be their enemies. They were not, in other words, a popular bunch, and only really had the support of other rich Athenians. Eventually, a coup led by exiled Athenians toppled the regime of the Thirty. The Thirty themselves were either killed or exiled, but most of their supporters were granted amnesty. Although the Thirty only reigned for a few months, their government traumatized the Athenians, not least because it demonstrated incontrovertibly that many of the rich would turn on the democracy, if given the chance.

What does this have to do with Socrates? Though Socrates' origins were rather humble (his father was a stonemason), he hobnobbed with some of the most powerful men in the city. Among these men was Critias, a close associate of Socrates who also happened to be one of the most bloody-minded among the Thirty. Another one of Socrates' students, Plato, was related to Critias by blood (his mother was Critias cousin). In Plato's version of Socrates' defense speech, Socrates claims that the Thirty asked him to bring a man they wanted executed to them, and that he refused to do so. Whether this in fact happened or not, the point is that Socrates had a close connection to the Thirty, and that this was a mark against him (though because of the amnesty, no one could prosecute him for anything relating to the Thirty).

Then there is the question of what Socrates spends his days doing. We think of Socrates as a philosopher, but that term was not as well defined in the early fourth century as it is today, and certainly wasn't a profession in the eyes of most. What Socrates seemed to be was a sophist, a teacher of rhetoric and oratory of the sort rich men hired to tutor their sons. Many Athenians had a deep mistrust of sophists. Because Athens was a direct democracy--in which every citizen could go to the assembly, hear speeches, and then vote on a course of action themselves--the worry was that sophists, with all of their rhetorical tricks, could make the rich into demagogues who would be able to manipulate the assembly for their own private gain. So Socrates looks like a sophist, and this is another strike against him.

Finally, the issue of Socratic dialogue. Today, we think of Socratic dialogue as an important and useful teaching device. And it is, in the right context. But if you go around asking those minding their own business "what is justice?" or, "who is truly wise?" they're typically going to flounder about and not be very cogent. And if you do this questioning with a group of students in tow, all of whom are rich young men (poor young men are off working), then it begins to look like what you are doing is not trying to find truth, but rather to embarrass members of the people in front of an audience of would-be oligarchs. On this understanding, the purpose of the Socratic method is to egg on rich young men to seize power from the people by showing them that the people are not wise. He is, as his prosecutors put it, "corrupting the youth."

So Socrates does not die because he is a pro-Spartan activist and the Athenians couldn't handle a bit of free speech. He dies because he was associated with the most violent coup in the history of the democracy, because his profession seems to be one which undermines the mechanics of the democracy, and because his namesake method looks like an attempt to mock the wisdom of the people (the demos). The idea that the demos, as a collective, was wise was a cornerstone of Athenian democracy. In the fragile post-thirty peace, a jury of Athenians thought that Socrates posed so great a threat to that cornerstone that he had to die.