If the world feels a distaste over the Qatar regime, there is a pretty simple solution: stop buying their shit.
Until the 1950s, Qatar was a tribal country of 30-40k fishermen and illiterate sheepshaggers (well, camels perhaps) and you can dig like a moron to try find out if they had any laws or anyone carrying out punishments etc. When the petrol industry found good stuff in the desert, the UK moved some 50k people (English and Indian mostly) over there to work and the country slowly started to modernise - they no longer had to get shipments of water from Bahrain, electricity and air conditioning started to be a thing and so on.
Not that the emir gave much of a shit. He nodded to British proposals about building some schools and stuff and then went back into his palace to count the petrol money. The average Qatari national still seemed to be doing what they've always done - fishing, pearl hunting and playing with those nice camels.
The Brits left (at least their responsibilites; they did not mind keep cashing in), the Emir was thrown out by his son (or nephew or something) who thought he could make some more quid out of the whole oil thingie and invited the yanks and the saudis and whatnot to come to the party. The Qatari nationals going about their shit suited him well, more money for him if his people wasn't educated and demanding liberties and better living standards.
In the 1990s some of the Al Thani family thought "well its a little bit of a pity that our own people are stuck in the 1800's while the foreigners are building this country" so they threw out the Emir in 1995 or 1996 (or something like that, don't remember exactly). The new bloke was more development-oriented - they started to open up to the world. Both with some western ideas - somewhat 'free' press and some places where you could have a beer and stuff - and some ideas from neighbours who had successfully kept their power, such as the Saudis, meaning harsher penalties, tougher laws and actual execution of these laws.
This approach worked well for them. Population since 1995 is up from around 400k to 2.8m today, country is filled with skyskrapers and shit and not many are fishing in the ocean or shagging camels and the standards of living have skyrocketed.
That the new bloke taking over in the 90s liked to flog homosexuals and infidels and what not wasn't seen as a problem by anyone as long as Qatar was a quiet desert country where you could make oil/gas and money. Now when they want political influence that represents their economical importance, people think it is a problem. Our states & governments however clearly doesn't give a shit: as long as Qatar is providing them with stuff, they can flog gay people night and day.
If our governments and corporations didn't feel that way, and boycotted Qatari resources, Qatar would not have the competence or manpower to keep the country going - it would revert into a camelshagging fisherman country, and the locals would not have it (as they're accustomed to better life standards now) and would get rid of the Al Thani family for good.
The bloke taking over in 2013 (or something) also was a big fan of the whole gay-flogging stuff but less so with the free press stuff his predecessor thought would be nice... and no one except for liberal powerless westerners give a shit.
The problem is not the Qatari regime itself, but those who make it successful and want to retain the status quo because it is profitable.
This includes: the Qatari nationals (great improvement in standards and finances), the west (plenty of American and other western oil companies profiting), the Asians (Japan, China, Korea etc. are the main ones importing from Qatar), Russia (exporting stuff to Qatar), Turkey (exchanging military stuff and funding some terrorism together)... the only ones a bit grumpy are, occasionally, the neighbouring states.
Are we supposed to sit and mutter about the immoral Qataris in front of the televisions while our states and big business are doing the exact opposite thing?
I think if we want to change things in Qatar, we probably have to start questioning our own countries and businesses dealing with Qatar rather than to "show their regime" how unhappy we are about their lack of morals.
