"Since I am the only guy with Linux experience I would have to support the Linux installations. Now the problem is what works perfectly fine for me may be a horrible experience for some of my coworkers, and even if they would only be using Firefox, Thunderbird and LibreOffice I don't know if I could seriously recommend using Linux as a desktop OS in a business. Instead I want to set up one test machine for users to try it and ask THEM if they like it. The test machine should be as easy and painless to use as possible and not look too different compared to Windows. Which distro and what configuration should I choose for this demo box?"
What you are describing is an impromptu usability test. And that's a good thing to do, especially if you are planning to recommend a particular desktop environment.
But what you need to start with is applications. Running Linux on the desktop is great (I do it at work and at home) but if you have users who need to run Photoshop, or a Windows IDE, or some particular finance application, it's going to be awfully hard to do that on Linux. But let's say your organization has all your applications in the Cloud or on an internally-hosted web application server, and these web applications run fine in Chrome or Firefox. That's a different story. But my guess is that you'll have at least a few programs that require running on the desktop.
My recommendation would be to find interested groups who'd like to try Linux on the desktop, and start there. Make it a pilot project. Take it slow, and meet with someone from that group daily to make sure you're addressing any pain points that come up. Things you'll want to watch out for are shared storage (like on a file server) and printing.