Upgrade to an unsigned version of iOS - possible or not?
No. It is not possible to installed an unsigned version of iOS.

The only version you can upgrade your iPhone 5s to is the current latest available for the device which is iOS 12.2.

If you have the source code for the app that you wish to test on iOS 11, you can test the same by building it for iPhone 5s Simulator running iOS 11 (you can test the software based features but not all of the hardware based features this way).