[weak]

Any idea, what could cause that issue? Any help would be truly appreciated.

That's because the fire tv currently doesn't support de-interlacing. Programs like ZDF (720p) will be fine, but Sky (1080i) won't.
Unfortunately there's no solution for this issue as of now, and it's likely this won't change in the foreseeable future.

Take a look at this thread for details.

[hello1893]

Very very interesting! I'm grateful you are letting me know. Can you explain to me, why I recognize these "errors" only in 1080-soccer but not in 1080-movies?

Second question: Would Kodi installed on a RaspberryPi have the same (or other) errors while watching 1080-materials with full Dolby capability?

Alternatively, I guess I could run Kodi on my MacBook and have it connected via an HDMI cable to my second TV. Would you agree this should work? Any other recommended workaround solution outside AFTV?

Again, thank you so much for letting me know!

[weak]

Your movies are probably 1080p (progressive), while soccer is broadcasted in 1080i (interlaced). But even 1080i movies tend to look better, as the interlacing effects are more visible in fast moving scenes like sports. For a detailed technical explanation take a look here.

Regarding the RasPi, I don't really know if they are fast enough to de-interlace 1080i. They might also support interlaced output, in that case the TV would do the de-interlacing (maybe someone else can answer that).

You shouldn't have any problems using your macbook.

If you're looking for a smaller form factor with passive cooling, I guess a NUC would fit the bill. But you probably have to spend at least 300 € ...

[Koying]

Re interlacing, please give http://forum.kodi.tv/showthread.php?tid=213600 a try