I need to sort several thousands of pictures. To do that quickly, I am looking for a tool to select the pictures I want and then copy these in a new folder in an efficient way, ideally just using the keyboard as I browse through the files in a fullscreen mode. Are there free software that can do that?
I would use https://www.darktable.org/ - It's in many ways similar to Adobe Lightroom, arguably (i.e. imo) darktable is better.
This is actually a raw converter and does much much more than you need to do.
To solve your problem I would:
1) create a 'collection' of the images that you need to sort through.
2) Delete all images from the collection (default keyboard shortcut should be the delete key, see here: https://www.darktable.org/usermanual/en/shortcuts.html)
3) Export the collection, as needed.
If you don't know about darktable and are intimidated, I recommend diving in anyway. This tool has many many uses and learning more about it will be usefull.
Besides Darktable @Martin mentioned, if you use Windows you can try IrfaniView. It has two modules, a single image viewer and a thumnail one.
In the single image view the one you are watching is the one selected... duh, and you can move it into another folder.
On the thumnail module you can click on many images to move them.
If you are really sorting photos, MOVE them, do not copy them.