Do people in soul society remember their life in the "living" world? So we've seen that souls (ghost) in the living world retain their memories, even if they've turned into a hollow. However, do those memories get purged when they are sent to soul society?

The reason I ask is because I'm wondering what would happen when Ichigo died? Conjecture - he gets integrated right into a squad, new squad made with him as captain, etc, etc... Since he's basically qualified as a shinigami through trial by fire anyway.
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.