Short answers, pulled from the story.
Charles Darwin popularised the term, contrasting it with artificial selection, which is intentional. Darwin actually came to regret the phrase and told Charles Lyell in a September 1860 letter that he preferred "Natural Preservation" instead.
No. Alfred Russel Wallace independently conceived the same principle and sent Darwin an essay describing it. In July 1858, both men's work was presented jointly to the Linnean Society of London, establishing co-discovery. Darwin acknowledged that William Charles Wells in 1813 and Patrick Matthew in 1831 had proposed similar basic ideas, though neither had developed or evidenced them.
Fitness is not strength or longevity but reproductive success. An organism that lives half as long as others but leaves twice as many surviving offspring has greater fitness. Because chance affects individual outcomes, fitness is technically an average measured across all individuals sharing a genotype.
"Survival of the fittest" describes the removal of less fit variants from a population through differential reproduction. J. B. S. Haldane called this process substitution. "Improvement in fitness" is different: it depends on a variant accumulating enough replications for a beneficial new mutation to appear, regardless of competition. The Kishony Mega-plate experiment demonstrated improvement in fitness directly.
Darwin's theory depended on heredity but had no valid hereditary mechanism. Gregor Mendel's work on inheritance was rediscovered in 1900, long after Darwin died. The modern synthesis of the mid-20th century united Darwinian evolution with Mendelian genetics. Ronald Fisher provided the mathematical framework in 1930; Theodosius Dobzhansky showed in 1937 that mutation supplies the raw material for selection rather than competing with it.