I'm having this same exact problem. It just started out of the blue today and I'm attributing it to that new Windows 10 Creative update that they installed. The console is there, it's just invisible, you can literally type commands and hit enter and they will work and then hit F4 again and it will exit out of the invisible command window. This is manageable for me, but...
The main issue is that when I go to a station and go on duty, it will load LSPDFR like normal and then out of nowhere just terminate. It's definitely a Rage issue IMO because of this. It won't even make that chime it makes when LSPDFR crashes, it just silently terminates all my plugins.
I'm going to try and re-install RageHook and see if that fixes it...