What are Babylonian and Assyrian?
They are languages which were spoken in Ancient Mesopotamia (modern Iraq). They belong to the Semitic family. Many scholars prefer to think of Babylonian and Assyrian as two dialects of a single language ('Akkadian') rather than as two separate languages.
When did they die out?
Hard to say exactly. As spoken languages, probably around 500 BC (some scholars would say later), but Babylonian continued to be used as a scholarly and liturgical language (like Latin) at least as late as the first century AD.
Given they are dead, how can one tell how Babylonian and Assyrian were pronounced?
The main things to work with are: comparison with related languages (e.g. Arabic, Hebrew, Ethiopic); sources in which Babylonian and Assyrian words are written in the Greek or Aramaic alphabets; sources in which Greek and other foreign words are written in Babylonian and Assyrian script; patterns within the spellings of Babylonian and Assyrian words.
Much about their pronunciations is still being discovered, though for obvious reasons much else is lost forever.
Who are the readers in this website?
They are all scholars of Ancient Mesopotamia.
Why do different readers sometimes read different versions of the same composition?
The readers might be following different manuscripts (i.e. cuneiform tablets), or making different judgements about how an ancient spelling should be parsed.
How much agreement is there among specialists over how to pronounce Babylonian and Assyrian?
Listen to the recordings and judge for yourself ...
Did Babylonian and Assyrian poetry have rhyme and metre, like English poetry?
Rhyme, no (at least not to the same extent as in English). Metre is debated. Some scholars think that the penultimate syllable of a poetic line was heavy (i.e. long or stressed, or both).
Where can I read more Babylonian and Assyrian poetry and literature?
Try the books by Stephanie Dalley (Myths from Mesopotamia Creation, The Flood, Gilgamesh, and Others), B. R. Foster (Before the Muses An Anthology of Akkadian Literature) and A. R. George (The Epic of Gilgamesh). Online, see the eTACT project.
And what if I actually want to learn Babylonian and Assyrian?