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. For about 3 minutes it is possible to run in the background but after this period, the app is suspended by iOS. System Status uses the 3 minute window to record CPU, memory, battery, disk and data usage for the graph view.

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