I know an Einstein-De Sitter universe is a "flat" universe, i.e. with a K ("bendiness", sorry, don't really know what to call it in English) factor of 0, but does that entail anything as to its contents? The Wikipedia page is quite lackluster on this aspect, and my astronomy course textbook merely slightly less so (it says it's made up mainly of "non-relativistic, dust-like matter").
Bendiness = space-time curvature.
And yes, it is flat (k=0) for the EdS model. Essentially it is the epoch at which only matter is thought to dominate (no contributions from radiative pressure or dark energy, $\Lambda$).
It is only a model. The EdS universe is thought of as a static Universe which is expanding indefinitely. It just makes solving certain equations in Cosmology a little easier, but it was perhaps due to Einstein's stubbornness in accepting a dynamic Universe.