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: 
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.
The best tool for this is process explorer by Windows Sysinternals. It's free tools that can view file and registry handles of any program running on the system.  

Here's the link:
<https://docs.microsoft.com/en-us/sysinternals/downloads/process-utilities>)  

[Here's an image of the open handles of Explorer.exe](https://i.stack.imgur.com/t1lEx.png)