How flat was the world? Here's an old map from the Ambarkanta of the 1930s demonstrating exactly how flat the world was - which was actually quite curved indeed.
Of course, we're talking about a strictly mythical period in Middle-earth's history here, so normal laws of physics shouldn't be considered to apply.
As to whether or not one could see vast distances, the main reference is the (supposed) ability to see Eressea from Numenor, which is mentioned in the Akallabeth:
...at times, when all the air was clear and the sun was in the east, they would look out and descry far off in the west a city white-shining on a distant shore, and a great harbour and a tower.
For in those days the Numenoreans were far-sighted; yet even so it was only the keenest eyes among them that could see this vision, from the Meneltarma, maybe, or from some tall ship that lay off their western coast as far as it was lawful for them to go.
Of course, Tolkien doesn't actually say anything about how far this distance was, so it's difficult to form a final judgement based on this passage.
The words "when all the air was clear" are relevant, however, because the distance one can see is not just dependent on the distance to the horizon, but also on clarity of the air, as Richard mentions in his answer.