There was a case a few years ago - I think it was in DOH actually - where airside security busted someone with drugs - a random search IIRC. Marched straight off and charged, as if landside. I mean, no-one would think that in the middle east it would be otherwise?
There was also one of those 'Airport' shows - Dubai I think - where they showed the operation of security airside, searching people etc. Again, clear that Emirati law prevailed.
Absolutely no doubt in my mind that once off the plane you are subject to the laws of the country you are surrounded by. In the air its less clear, but if there is a crime in the air over a country and the plane lands in that country, then local law and local law enforcers will be 'persuasive' I think.
Now, going back to the incident in question, it may be that this was illegal under Qatari law (you would hope so, but can't be sure). But I dare say that there will be nothing done beyond what we have seen already unless there is widespread international pressure, which is why I keep looking for evidence that other nations are doing as we are. Think of when the Saudis murdered someone within their embassy in Turkey. No question, Saudi territory. Yet there was at least some repercussions for the perpetrators, if no accountability to the top.