No. A quadrant is an entirely different instrument.
True. There was zero doubt at that time among educated people that the world was round. Zero.
The question was the size and geographic distribution. The size was also known with a certain aproximation. The latitude difference between known geographical points was known with a reasonable accuracy and so the length of the degree was also known with a certain (rough) accuracy. Some would go with higher figures and some with lower. Columbus, like many people, not being immune to adjusting the facts to suit his purposes, proposed a very small world.
Another variable was the distance between Spain and Japan by way of the Middle East. No one knew with any certainty how many degrees of longitude that might cover and Columbus proposed high figures so that the remainder, the distance from Spain to japan going west would be small.
How much of that he really believed and how much of that was market8ing we do not know. probably there was a combination of all of it. He hd probably heard stories about land having been sighted to the NW Atlantic. Whether that was true or myth we do not know but he was probably influenced by it. He then tweaked the figures of longitude so they would fit his pre-concieved notion of where this land might be. And he assumed this land was Japan because no one had ever heard of anything else. He devised a selling plan which worked because it sold but we don’t know how much he really believed.
Note that he also constantly lied to his crews about the distance being made good so that they believed they were not as far from Spain as they really were and so they believed they were closer and easier to get back home.
One thing is clear and that is tha Columbus knew very well the currents and winds of the Atlantic and navigated mainly by dead reckoning rather than by astronav.
But, yes, everyone knew the earth was round and no one really had any fear of falling off the edge. Their fears were much more real: in those times life at sea was tremendously unsafe and many who went never came back. But life inland was also very hard and people took risks that were unthinkable today.
Well into the 20th century life at sea was very risky and men were often lost overboard without much thought. Read any 19th century story about sailships and you will see what I mean.
In the National Geographic Magazine, in the February 1931 issue, there is an article by Alan Villiers titled “Rounding Cape Horn in a Windjammer”. You can find it online.
In 1893 the Spanish sailship “Nautilus” was sailing around the world when a sailor fell from the rigging and lost his life when he hit the deck.
In “Two Years Before the Mast” Dana writes:
These things were part of everyday life and they were accepted as such. Only in very modern times do we expect complete safety. And this only in more advanced countries. There are still many hundreds of thousands who would assume the risk of drowning or otherwise dying in exchange for a small chance of reaching a developed country where they will have better life prospects. Many are dying every day in this pursuit.