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So the mess in Syria is our fault because we toppled the Shah in Iran?"

Partially, if you consider Hezbollah and Iranian involvement (allied with Assad).
The mess in Syria is due to a second invasion of Iraq without any reason. The very-ill conceived invasion, followed by a absolute disaster in the immediate post-war policy (like making 2 millions unemployed but armed former soldiers, firing of all the administration because they were previously in Saddam's party -as if they had a choice), the absolute ignorance of the local reality created a vacuum of power and corruption, even bigger than before, spreading poverty in the population, pulverisation of the Iraqi society and return to old structures (tribal and religious).
All this created the destruction of the national identity, and this lead to the collapse of the Iraqi Army (that no reason to fight for a non-existent country) as show with the collapse of this army facing IS.
This collapse allowed the creation of a territory and the seizure of natural resources which will help IS to control more territory and population.
And because IS is ideologically close to Saudis, and they are mostly Arabs and Sunnites, they went in direct confrontation with Iran/Shiites, and of course, the Syrian Alawites/ Assad, Kurds. Because they are in confrontation with the Kurds, Turkey decided to trade with IS and to attack the Kurds in the back, having themselves a problem with Kurdish nationalism. They were inching for this, as a autonomous region in Iraq was not for their taste. And, at the same time, they were helping the Turkish populations in Syria with the long term aim of an annexation, when Assad Regime would have fall.
USA and Europe turned a blind eye on this, as they wanted Assad regime to fall so Russian base (the only one) in Mediterranean Sea would have to be closed. And Turkey is part of NATO.
Arab regimes did as well, because the "moderate" opposition chosen by the Western Countries were theirs, as ethnicities and ideology. The added value to fight a secular regime was too much to resist anyway.
Of course, if I can see it, so Putin, so he run to Assad's help, and after massive air support, Assad started to regain some territory.
I still do not understand why Turkey decided to shot down a Russian plane, but it didn't add the desired effect. No only Putin didn't retaliate but he pushed harder and the political landscape did shift in his favour (and Assad).
I am quite sure than some will come with explanations that he is in fact more isolated than ever, but Russia is now fully back on the political agenda after Crimea.
So, to answer your question, yes, our policy of Regime Change is responsible for the mess in Syria. Directly.
In ignoring the deep streams under the surface of the region, focusing only on the religious (and not completely grasped), ignoring the old war between Persians, Arabs, Kurds, diversity of societies and the political landscape provided by the mixture of all these elements, we are directly responsible for the mess in the region.
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