If I want logging for the "internet" facing context on an ASA, do I have to configure logging on that context, or will the logging on the admin or system context also send logs for the other context?
Logging must be configured separately in each "customer" (non-system or -admin) context that you want to receive syslog messages from regarding its activity.
The admin context can send syslog messages related to its own and the system context status.