The English word “sleep”, which serves as both a noun and a verb, is Germanic in origin, probably originally derived from the Gothic word sleps, and closely related to the modern German Schlaf and the Dutch slaap. The words “slumber” and “doze” are also Germanic in origin, and “drowsy” is from Old English (and so also Germanic in its ultimate roots).
Most European Romance languages take their words for sleep from the Latin somnus (e.g. sommeil in French, sonno in Italian, sueño in Spanish, sono in Portuguese, etc), which is also the root for English words like “somnolent”, “insomnia”, “somnambulist”, etc. Many other languages also have a related root, e.g. Swedish sömn, Danish sevn, Russian son, Polish sen, Bulgarian sun, Serbo-Croatian san, etc.
Sleep-related English words like “soporific” come from the Latin sopor (meaning deep sleep, but also the name of the Roman god of sleep), and words like “hypnotic” and “hypnogenic” are from the Greek hypnos (meaning sleep, but also the Greek god of sleep). “Dormant” comes from the same roots as the French verb dormir, namely the Latin verb dormire, meaning to sleep.
The opposite of sleep as a noun is usually referred to as “wakefulness”, a rather cumbersome word derived (like “wakeful”, “waken”, “awake”, “waking”, “awakening” etc) from the verb “wake”, which is from Old English and ultimately Germanic roots. Incidentally, “wakefulness” refers to the alert state of mind normally experienced during the daytime; “sleeplessness”, on the other hand, has slightly different connotations and usually refers to an inability to sleep during times normally reserved for sleeping.
The etymology of the word “dream” is less clear. It appears to have descended from an Anglo-Saxon word dream meaning joy, merriment or music, or possibly from the West Germanic word draugmas, meaning deception or illusion. But it is interesting to note that the Proto-Indo-European word for sleep is thought to have been dre- or drem (Proto-Indo-European is the ancestral root language behind most European languages, as well as many other languages in Eastern Europe, the Middle East and the Indian subcontinent).