You are right, El Campesino was (nominally) a communist, but he was opposed to the stalinists and he was more an anarchist than a communist. It is doubtful, but he claimed to be abandoned by communist forces when he was encircled in Teruel. About his anarchist past:
http://www.fundanin.org/gorkin7.htm (in Spanish, sorry)
Some Republicans considered themselves communist and the vast majority of International brigades were communist.
Right. But the PCE (Spanish Communist Party) has little influence. The dominant party was PSOE (Spanish Socialist Worker Party). I agree that, by current standards, it would be considered a communist party.
Many Nationalists considered themselves fascists, most notably the Falange.
The founder of Falange (José Antonio Primo de Rivera) didn't consider himself as a fascist. Franco was a conservative pragmatical with no ideology. Both of them said his movement had some similarities with Nazis and (notably) Italian Fascists. The same as republicans the war increased more and more the sympathy for fascism and communism, respectively.
As a whole and letting aside nuances, you could consider Falange as a fascist party, at least until 1936. During the war became the only party in nationalist side and many rightist with no ideology joined the party.
I agree with you that the Republican forces were often incompetent but I would say this is more because of internal squabbles and soviet meddling
I disagree. Although Russian influence was negative when speaking of internal quarrels, they were the most organised force in Spain.
When the war began the republicans were notably better in weapons but they were so badly organised that they spoiled this advantage. Only communists enphasized organization and unity (under its lead, of course).