Do Orcs and Uruks sleep or do they just remain awake 24/7 like Sauron does?
Yes: they need some sleep, but they're trained to be hardy.
Orcs are mortal beings after all, and it seems reasonable to assume that they do need sleep just like Men, hobbits, etc. This is confirmed by the following quote (which I found in this excellent answer to another question) from text 10, Morgoth's Ring, of History of Middle-Earth (emphasis mine):
They needed food and drink, and rest, though many were by training as tough as Dwarves in enduring hardship. [...] They died and were not immortal, even according to the manner of the Quendi.
Exemplifying the part about enduring hardship, consider the Orcs who kidnapped Merry and Pippin, who apparently ran without ceasing for 3 days solid. They are never described as stopping to sleep in the chapter (The Two Towers chapter 3, The Uruk-Hai) devoted to them and their prisoners. Their toughness and tirelessness is also touched upon by the Three Hunters in the preceding chapters:
"But [the Orcs] go with a great speed for all that," said Aragorn, "and they do not tire. And later we may have to search for our path in hard bare lands."
-- The Two Towers, chapter 1: The Departure of Boromir
"Unless our enemies rest also, they will leave us far behind, if we stay to sleep," said Legolas.
"Surely even Orcs must pause on the march?" said Gimli.
"Seldom will Orcs journey in the open under the sun, yet these have done so," said Legolas. "Certainly they will not rest by night."
-- The Two Towers, chapter 2: The Riders of Rohan
Unlikely. Since elves don't sleep (or at least not like humans do), it would stand to reason that orcs don't either.
There are suggestions in the Silmarillion that the orc species originated with elves that were kidnapped by the forces of Morgoth and then experimented on. Tolkien had to introduce an explanation like this because only Eru Illuvatar can create intelligent life of any kind, and he didn't directly create orcs.