Question: Is there a way to change the visual keyboard-theme via C#?
Or at least, some way to edit its properties, like background color etc?

Answer: 
Nope, this is one of those things that cannot be changed by third party applications short of various hacks (which would not be allowed in the marketplace). In general if something is controlled by a user setting then an application will have no way of changing it. You can detect whether the user has selected the light or dark theme and change the colours in your application accordingly, but your application cannot change system settings.
Try changing ApplicationBar or maybe Systray background and foreground colors. I saw an app that had ApplicationBar and Systray set to white background and back foreground and the keyboard was black on white even with the dark theme. I think there may be a correlation.