Do people in soul society remember their life in the "living" world?
Yes, they do. In Episode 4, we see a soul of a boy trapped in a parakeet. Later in the series, when Ichigo, Chad, Uryuu and Inoue enter Soul Society, they meet with him again (in human form), and he remembers them.

It is unclear whether or not they lose their memory after long spans of time spent on Soul Society.