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.
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)