How do I choose the license for the actual github repository?
Legally, you can only choose the license for your code that you author. If you have dependencies, do not host that code in your github repo. So the license you have chosen is for just your code.
Use submodules to download the other code. Note somewhere in your project that your project uses code from other repos with different licenses.
Is there a tool or a webpage to answer that question? Or to give a list of possible licenses for my github repo dependent on the used packages?
@Z Z found a tool in comments. I can't address how well this works. I tend to do this step manually by reading each license personally.
An example of how to enumerate the licenses in a safe manner is the FreeBSD port for cups-filters. The distribution for this package has 8 separate distinct licenses.
BSD4CLAUSE GPLv2 GPLv2+ GPLv3 GPLv3+ LGPL20 LGPL21+ MIT
Each individual package has the potential to have a different license. All the licenses are shipped with the package but the submodules are downloaded at compile time.