Hellenistic mean "Greek-like); it refers to the period after the death of Alexander in 323 BCE until the last Greek monarch, Cleopatra, lost Egypt to Rome in 31 BCE.

Plenty of people, Barry Strauss for instance, think the Trojan War is an historical event. It's a question of how well you judge oral poetry is at transmitting information. Judging from his book, Strauss has a very high opinion of the fidelity of oral poetry. There's a burn layer at Troy VII some people associate with a Greek invasion; skeptics only admit there's a burn layer and claim ignorance of who did the burning.

As for the fidelity of oral poetry, I have a couple of problems. The plot contains a lot of elements of IE mythology; you can find parallels to the Skyros story in Irish myth easily. The plot also seems to me to have been rearranged; the Achaeans were there for ten years before they thought to build a wall? That should have been one of the first things they did. Also, Nagy (The Best of the Achaeans) demonstrated pretty well the Homeric epics operated in a world populated with other, related epics which you need to understand Homer properly (things like the use of duals in the embassy scene, when the embassy to Achilles in fact was made up of three).

That doesn't mean the war didn't happen, just that I don't think Homer is good evidence for it.

For the Peloponnesian War, yes that happened. We've got so much stuff from that period it's silly to doubt. Aside from Thucydides (a general in the war), we've got contemporary plays that address issues (the Lysistrata, for instance), other historians, reports of things like trophies from the war hanging around Athens decades later.... it really isn't a problem.