The selling of duty free alcohol and gambling happens once you go through immigration thus you are 'air-side' and in international airspace. You have been through immigration and are thus no longer in Qatar, Qatari and Islamic laws no longer apply.

When returning to Qatar you you will notice no gambling or alcohol on sale once you go through passport control. The laws of Qatar kick in then and you cant bring in booze.

So no, the airport is not muslim, in fact its not even Qatar once airside.thought everyonr knew this?

I don't go to mythical places with strange men.
-- Douglas Adams, The Long Dark Tea-Time of the Soul.