We're spending a week on the SlingBox location. First the good news: it works!
There's no real bad news, besides that I can't really say how our problem was solved.
I had a look at how all devices were connected, and the only thing I did was disconnecting the antenna cable, since the SlingBox was already connected with the cable box through a scart/composite cable. I then powerd up both cable box and SlingBox, went to the pc and the programm suddenly found the box. Everything else went as it should.
Now, a month passed between today and our second failed attempt, which was the reason for this topic. My parents-in-law had disconnected everything then, and didn't do anything since. The only noteworthy event was an internet outage after which the cable company had to remotely 'fix' something, after which the internet worked again. Not sure if today's third succesful attempt was succesful due to something the cable company did.
Anyway, I'm just listing this here as a reference. I appreciate the help I received, even though I had no chance to use the advice. Giving feedback is the only thing I can do to repay
Next week we'll see if we can connect remotely as well.