Usually, but they did mention Jervis Tetch as the Mad Hatter's real name, and Mr. Freeze was given the name Dr. Shivel in his first appearance. Then there was Siren/Lorelei Circe, though I was never sure if that was meant to be her real name or an alias. King Tut's real name was revealed in season 3 as William Omaha McElroy.
No, by 1966, the majority of them did. Penguin's name was revealed as Oswald Chesterfield Cobblepot in a 1946 newspaper comic strip storyline. The Riddler was Edward Nigma from his debut in 1948. Catwoman was revealed to be Selina Kyle in 1951.
Also, 2/3 of Batman's villains were created for the show. Only nine, I think, came from the comics: Joker, Riddler, Penguin, Catwoman, Mr. Freeze (though he was Mister Zero in his sole previous comics appearance), Mad Hatter, Clock King, False-Face, and Eivol Ekdal from "Zelda the Great"/"A Death Worse than Fate." Of those, only Joker, Mister Zero, False-Face, and Clock King were still anonymous as of 1966 (though Clock King, actually a Green Arrow foe in the comics, was identified as William Tockman in 1979).
So the TV show's avoidance of real names, and its treatment of the rogues' nicknames as their legal names, was its own comedy conceit. It wasn't just villains; Commissioner Gordon was never called by his first name on the show even though it had been established in 1951, and Chief O'Hara, created for the show, had no first name at all (though he was later dubbed Clancy O'Hara in the comics).
On the other hand, Alfred didn't get the last name Pennyworth in the comics until 1969, although some early '40s issues had called him Alfred Beagle or even Alfred Jarvis.
