is it possible using my old operative system macOS sierra 10.12.6 on a brand new mac mini?

The drivers hidden in macOS that add support for the various hardware components making up a Mac are only provided by Apple. Apple only provide new drivers in newer versions of macOS and therefore in order to support newer hardware you also need a newer matching version of macOS.

This is different to Windows where the drivers are separate to the operating system and usually not even written by Microsoft. Hence it is much easier to installer older versions of Windows on newer hardware with newer drivers.

There is however one way you can run an older version of macOS on a new Mac, this is to run a copy of macOS inside a virtual machine. The three main virtualisation programs are -

1. Parallels Desktop
2. VMware Fusion
3. Oracle Virtualbox

All three are able to run virtual copies of macOS on a Mac and this is legal. Doing so or trying to do so on a non-Mac is not legal.

VirtualBox is free but delivers a much less pleasant experience for running a virtual Mac. If the purpose is to just be able to run some old programs not available for Catalina then it may be 'good enough'.

No, have a look at > https://support.apple.com/en-us/HT201686

No. A 2018 Mac mini won't run anything older than Mojave (Mac OS 10.14), and a new one that comes with Catalina (10.15) installed may not even let you run Mojave.