The 1828 book 'A History of the Life and Voyages of Christopher Columbus' by Washington Irving contained the 'Columbus proved the earth was round' myth. This book was very popular and likely contributed significantly to the spread of this belief.

I'm not sure about the development of his memory in Europe or Latin America, but in the USA, he had long been celebrated, but if we're talking about official celebration, then Columbus Day was created in the early 20th century as a result of the lobbying of Italian American organisations (as Columbus hailed from Genoa) and the Catholic Knights of Columbus group.

I know it is quite late, but this old answer about Columbus Day would help answer your second question.

Notion* sorry