No. In the Italian sentence sleep is a verb (to sleep) and in yours it is an adjective (asleep). They don't mean the same thing at all in English or Italian.
You can say we sleep badly, but you cannot say we are asleep badly.
Dcarl1 is absolutely right: "We are asleep" is translatable with "Noi siamo addormentati" while "We sleep" with "Noi dormiamo." Then we have "Nobody sleeps" ("Nessun dorma," but it is an aria from Turandot).
Not to be nitpicky or anything but Nessun dorma means "let nobody sleep": it's an order, not a description.
Exactly, but a subjunctive with an exhortative meaning. And that is expressed in English with the "let+verb" construction.
Probably it is better to translate it with "None shall sleep" http://www.metrolyrics.com/nessun-dorma-lyrics-luciano-pavarotti.html
We don't ever say we sleep. We're asleep or we're sleeping is what we say in the UK