Is it possible to monitor other apps while System Status is in the background (and e.g. log the result into a file)?

Only in a very limited way. The app can run in the background for about 3 minutes, but after this period, it is suspended by iOS. System Status uses this 3-minute window to record CPU, memory, battery, disk, and data usage for the graph view.

For more information about background execution, please see this article.