After searching for solutions for hours I finally found the reason why I cannot select other iOS versions in the Simulator:
It's Mavericks!
In this post in the Apple Developer Forums (Developer account needed to access the forum!) an Apple employee states:
The iOS 5 simulators do not work on OS X 10.9 (Mavericks).
and he says the reason for that is:
The iOS simulator does not duplicate some of the low-level parts of iOS. For example, it uses the host OS X's kernel instead of trying to virtualize the iOS kernel. The iOS simulator requires that these subsystems look sufficiently similar between the simulated iOS and the host OS X. When a new host OS X changes too much it can break old simulators. At that point somebody makes an engineering and cost decision to either update the old simulator or drop support for it.
So if you have installed Mavericks (and do not downgrade if that's even possible) you have no option to run the Simulator with iOS versions < 6.
(If you want to test your app in the Simulator with iOS 6 and 6.1 SDK proceed as the other replies and comments on this page suggest: Go to XCode > Preferences > Downloads and download the corresponding simulator version in the section "Components".)