This package does not contain static Qt libraries. The .a files are used to ease the dynamic linking. On Windows there are two ways to use dynamic libraries.
The first one is by calling LoadLibrary (or using QLibrary that wraps it) at runtime. It's not convenient because you need to resolve all the symbols manually this way - lots of pointless work with a huge library like Qt.
The other way (preferable and used here) is to link to a library that will load and resolve the shared library (dll) at runtime for you. The .a files you listed are just that. You link to them and you still need the dlls at runtime.
As for how you use them: When you create a project in Qt Creator , in your .pro file you add the modules you need in the QT variable, for example: QT += core gui widgets network. After that you just run qmake (Build->Run qmake). This handles all the linking to the right .a files for you.
To have a static build (without the dlls) of Qt you need to build it manually from source package. Be aware that there are licensing restrictions for the open-sourced version of Qt when linking statically.
I also tried this on my Win7 machine but never succeeded. In the end I did a local compile on my RPi2 of Qt5.5. It compiled for 1041 minutes but is working nice now. I followed this guide for the local build. Instead of 5.4.1 I just got the latest source.
Could also be an option for you.
I'm glad you got it working. For the future, I wouldn't recommend copying the dlls to the windows system directory. Check out this page for deploying on windows. In particular check out The Windows Deployment Tool section.
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