The iPad does not have to be awake to download apps, though really large ones will sometimes stop and have to be resumed later.
If the app is not showing on the Storage list, nor the iPad screens, try looking for it in the App Store. Check both the Updates page and the Purchased tab.
And when you say that iTunes shows it as downloaded, do you mean on the computer? That would not be the same as being on the iPad (though if you sync, it should rectify that).
There are settings that cause iTunes and the iPad to download automatically to the other; but those settings can be one way, both ways, or neither.
In iTunes check Preferences > Store and see if you've got auto syncing for App, Books, and Music turned on. Do the same thing in Settings > Store on the iPad. If both are turned on, then both will eventually get the app. If only one is turned don, then only that side will get automatic downloads when purchases happen on the other. Did that make sense?