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 [Process Monitor](https://docs.microsoft.com/en-us/sysinternals/downloads/procmon) for this purpose. RegMon and FileMon are merged into this and it can now be used to monitor process, registry and files.
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)