There's mixed views on the CNT's eventual decision to work with the government and take certain roles: Emma Goldbloom for example was roughly supportive arguing that circumstances demanded it. While there's no doubt they eventually took part in the wider government of Spain in Catalonia itself they basically abolished it.
No they didn't. The Catalan government was always around and powerful enough to give the Republicans a headache due to their separatist leanings. In fact, the most powerful faction in Catalonia was the Catalan Nationalists, not the Anarchists.
As above this did happen but with remarkably little influence on Catalonia in general and Barcelona in particular. I'd also question whether CNT became an organised political party itself during that period as opposed to essentially a well organised pressure group... simply having members in government doesn't seem to me to be enough to be a party in and of itself.
Having members in government who associate with each other ideologically in a structured way and organizing voters to get and keep such people in government? That there's a political party.
By my understanding was that they did... they implemented a fairly widescale project of setting up syndicates (with about 75% take up, especially in their industrial powerbase) and had real power and influence... leading to the crackdown during the Barcelona May Days. There's some debate as to whether it was a socialist libertarian/minarchist system or the syndical anarchism they publicly called for but again, within Catalonia itself and Barcelona in particular, their power came not from the State but from their own power base and the State was allowed to whither away. The real issue is that there are many accusations... of which many are almost certainly true... that this Spanish Revolution was basically forced on the general populace by the use of coercion by a relatively small cadre of dedicated hardliners... anarchism at the barrel of the gun.
The Anarchists did not suffer a 'crack down' on the Barcelona May Days. They were defeated by Communists. A crack down would have had to have come from the Republican Government which was uninvolved. Further, Anarchist Catalonia did not take up the whole of Catalonia but just the parts the Anarchists occupied. And, as you said, they imposed their own form of order.
I think part of this has to be seen in the political context. Both the Republican government and the Soviet supporters were at best wary of the anarchists with the Soviets being especially troubled... much as they had to Nestor Makhno's Black Army in the early 1920's... and at worst openly hostile (as during the May Days). This was at best an unholy alliance... if not for Franco then the Anarchists would have been raging against the Republicans and the enemy of my enemy is not necessarily a friend you wish to allow to grow strong. There was vast amounts of Bolshevik propaganda against the Anarchists and the Soviets basically refused to offer them any anid.
No, they would have been raging against the Communists. They were working with the Republicans as a political party before Franco was anyone more important than some random general. It was not an unholy alliance it was what they sought to do.
As above. While I'm not saying the Anarchists were exceptionally effective (although the Durruti Column had some successes) their eventual disbandment and integration owed much to a growing Soviet influence and their wish to prevent there being a distinct Anarchist fighting force to oppose them. All of the groups aligned against Franco had one eye on the post-war Spain... and all paid for it.
Yet they were considered even less effective than Republican units or other militias. Even relatively they failed to stack up. I couldn't say how much of this was just because, being anarchists, they often refused to obey orders though.
I think under the circumstances... against the trained military supplied and aided by both Italy and Germany with at best limited support from their allies if not being forced into running street battles (again I point to May Days)... they did remarkably well. Some reports list production actually going up in Catalonia at the time and it was one of the few even somewhat successful attempts to put together an anarchist society. The police had basically been disbanded, almost all workplaces had been collectivised and other hallmarks of syndical anarchism the CNT put forward were in place. The coercion accusations lessen the impact but overall I think you can point to Anarchist Catalonia of that time as a decent example that anarchism can at least somewhat work in practice.
They did not have limited support from their allies, they had a mutually hostile relationship. Mutually. In fact, one might even say the Anarchists started it by raiding Republican depots (violently at times) to steal supplies and arms when the Republican government considered telling them they weren't going to arm militias which openly declared they might flout Republican authority (and they did flout Republican authority, Anarchists were infamous for retreating or attacking regardless of what High Command told them they needed them to do). In fact, the Anarchists were remarkably violent, though not more so than any other radical political group such as the Communists, and that might be forgiven by war time conditions.
And the police had not been disbanded, they had been replaced by police units made up of 'workers' who were loyal to the Anarchists. Again, when this was challenged as biased, the Anarchists responded with violence.