1. Hutton and Lyell were the people that influenced Darwin's idea that life could change slowly over a period of time. Both James Hutton and Charles Lyell were geologists who argued that Earth’s surface was shaped by gradual geological processes and that Earth must be far older than most people believed.
2. Malthus was the person who influenced Darwin's idea that some offspring are better suited for survival than others. Malthus was an economist who argued that human populations grow faster than the resources they depend on. He explained that population growth would always overpower food supply growth, creating states of hunger, disease, and struggle. That stimulated Darwin who than considered that some of the competitors would be better equipped to survive and that those who were less able would die out, leaving only those with the more desirable traits.
3. "Traits acquired during an organisms lifetime can be passed down to its offspring" is the parts of Lamarck's theory of evolution that is no longer supported. Lamarck was the first scientist that proposed that species change over time, but his idea of the inheritance of acquired characteristics is now considered as incorrect (yet, because of the knowledge of epigenetics today, some of his ideas might be partly true).
4. Darwin concluded that humans can influence an offsprings trait, and this is how livestock breeders and farmers influence Darwin's idea. The process of human influence is called artificial selection and it is a process in which organisms evolve traits useful to humans because people select which individuals are allowed to reproduce and pass on their genes to successive generations.