iPhone 5s stuck at Apple logo and progress bar
As I wasnt here for a longer period of time, here is what did I do to solve this: 

I finally managed to boot the phone. Here is what I did.

After I've already tried on another computer, a customer brought me a desktop PC (before I had 2 laptops). I installed Win 8.1, previous laptops have win 10. So from this point of view, I gained hope.

A fresh installed OS, newly downloaded iTunes.  As next, I've connected the device to the PC which succesfully installed new Apple drivers (I was carefull at those events).

So I ran into DFU mode (power off > sleep/wake + home button > release sleep/wake button). It successfully detected it in 'recovery mode'.  I chose RESTORE. I had the notification for the new 9.1 (which usually wasn't here - except probably for the first time I ran the app on other PCs), And as next the Restore button greyed out. I thought it was 'frozen'. As I checked in the task manager, the iTunes process was using network, which was meaning it downloads the new large 9.1 iOS.

When it installed, everything was as usual. Finished the extracting, verifying...that it rebooted and disappeared from iTunes. With no other option left, I decided to let it install, even if for hours. BUt then I unplugged it and saw low battery.  When I began with charging, after few minutes a vibration with "Hello" screen appeared.

I would advice other who have the same issue to be careful at these steps.