There was no argument or implication beyond 'SD has
something to do with ToE and Darwin'.
Going beyond that starts to get complicated quickly, even defining Social Darwinism precisely is difficult, especially now it carries very heavy negative connotations that were not present at the time.
Yes. The term was created by Darwin's cousin who was strongly influenced by Darwin. Had he not been so influenced then we might have had a different word for 'eugenics' coined by someone else
Seriously though, Darwinism was probably
partially responsible for the popularity of eugenics (which has both positive and negative forms) due to its influence on intellectual culture and the perceived links between the issues.
Had Darwin never lived though such ideas would still have been promoted and popularised as they don't owe their intellectual genesis to any single figure and Darwin was hardly the only person proposing theories of evolution.
Let's say it offered (perceived) additional support to existing ideas and was used as an intellectual foundation for additional ideas in this area.