Question: Is there a way to *monitor* how an installer for Windows writes its own stuff inside the registry?

Is there also a way to monitor which files are written on the disk?

Answer: 
[Revo Uninstaller](https://www.revouninstaller.com/) will track an installation and show you all files and registry entries that are written during an installation.
Use a software that does Registry Snapshots like [RegShot2](https://www.portablefreeware.com/?id=2505). 

> 
> Regshot2 Unicode is a registry change and file system change detection
>  diff tool that takes snapshots, generates HTML reports, and creates
>  automatic REG undo/redo scripts. The program can save/load snapshots
>  to file for later use, while registry path and file path
>  inclusion/exclusion rules can be set to include/exclude paths in the
>  snapshots.
> 
> 
> 

* Do a registry snapshot
* install the software
* do a 2nd snapshot and let the software compare the 2 snapshots for diffs.

Now you see the new/altered registry settings.