@Reimar: Is there news about this inconvenience?

I have done all I can, including running in a VirtualBox VM (a bit slow for the actual gameplay, but works fine for video).
It's up to someone else to provide sufficient information to track down the cause. I know that sometimes the game does things that normally would cause a program crash but Windows compatibility mode catches it and makes it only go slow instead, but I don't know if that is the cause here, what triggers it or how to reproduce, i.e. absolutely nothing I can do currently.
As a small start (unlikely to be of any help on its own) this question is still unanswered:

It's slow-motion.

I had the same problem and it was an xbox one controller driver issue. The instructions in this thread fixed the very slow game for me.

https://www.reddit.com/r/GrandTheftAutoV_PC/comments/48me06/psa_latest_windows_10_update_breaks_keyboard/

I also have an Xbox One controller.

IT WORKS! IT WORKS! but the mousekey is not visible....

Use the game's launcher and enable softwarecursor in advanced options. There are other ways to do that, just do a forum search for "softwarecursor".

I can confidently say I'd never have figured that out. The idea that the XBox controller driver would make things go slow like that is quite absurd...
Great that it works! (minus the cursor thing, but that is expected and was obviously already a problem back when the game was developed since they have the option for it in the launcher)

The missing cursor is actually a feature not a bug ;-)
GOG removed the game's original cursor when they added their ddraw wrapper (which comes with it's own cursor) so that there aren't two on screen. (This is still the case with the Steam versions which use the same wrapper. http://steamcommunity.com/app/355250/discussions/0/620712364023847868 )

I still have GOG's initial releases on my hd and they show the cursor just fine. When I redownload the games it's missing.

This was originally the reason for the software cursor option:

E. Software Cursor Mode

On certain video cards, the mouse cursor may appear
discolored or corrupt. The "Use Software Cursor"
checkbox under "Advanced Options" will fix this,
slightly decreasing framerate in areas other than
flight.