Latitude was somewhat easy to determine since antiquity and the equipment to find it was common on most ships as was current speed in the ocean. Most cargo ships sailing out of sight of shore would try to aim laterally for the city in question then sail near the coast to get to where they needed to go.

Longitude took until the late 18th century to find as it required an accurate clock set to a known meridian. Before then, it was a guess based on approximate recorded speed.

Columbus was an idiot who miscalculated the circumference of the Earth, but crews at the time commonly sailed out of sight of shore.

https://en.wikipedia.org/wiki/Celestial_navigation

That should answer all your navigation questions.

Regarding crew morale, I haven't studied Columbus since maybe HS, but I do recall the voyage was a fairly close thing. His calculations severely underestimated the size of the Earth, so he was sailing for much longer than he expected to be. I'm not sure if he had to put down any attempted mutinies, but it wouldn't surprise me.

how did Columbus stumble upon America without getting lost?

He was lost, he was sailing for the Orient. He grossly misunderestimated the circumference of the world.