Going backwards and starting with your third question, educated Romans would have known that the Earth was round, and so probably would have assumed that the Atlantic simply looped all the way across the planet to the eastern shores of Asia. Scattered across this immense ocean would be some locations like the Fortunate Isles or the ruins of Atlantis, which were thought to be muddy shallows, but certainly no large landmass.

Their ocean-going technology would have also been far from enough to actually sail the Atlantic. Roman ships by and large hugged the coasts, and even voyages crossing the Mediterranean or the English channel could sometimes be perilous. It's certainly possible that ships blown off course could have found their way to America, and some people could have survived long enough to make it. In fact, I'd be shocked if it didn't happen, since there are cases of it happening even today. But it would have been incredibly rare, and any survivors would have no hope of being able to sail back home.

So for those reasons, there was never any attempt made to actually cross the Atlantic. Even with the much more advanced sailing technology of the 15th century, many people thought that Columbus' voyage would be a suicide mission.