Sad I missed the recent panel AMA! I'm curious to know what happened when Nationalists captured territory from the Republicans, and what it was like to live in that territory while the war went on. Did they purge the populace of suspected leftists? Did they immediately replace the local government with something entirely new, or just replace local officials within the established system, or leave it intact under martial law? Was capture and occupation different for cities and towns than it was for rural areas? Thanks!
Did they purge the populace of suspected leftists?
This is the area I have studied the most out of your questions, so I will mostly confine my answers to this subject.
Bluntly, yes, there was a purge of virtually anyone with ties that were deemed to be leftist--for a total of 150,000 noncombatants murdered during the war. If you were an elected official, journalist, academic, artist, union member, leftist political party member, or merely being insufficiently ardent in your support of the Nationalists you could find yourself murdered in short order. Imprisonment was another likelihood, often in appalling conditions. Being forcibly dosed with castor oil so that you would soil yourself in public was another punishment. Even if you were entirely innocent (and I use that word hesitatingly, as what acts the Nationalists considered to be crimes was arbitrary and not based on the law) you could find yourself denounced by one of your neighbors and find yourself punished in any of the above manners. If your right shoulder showed a bruise when the Nationalists examined you, your execution was a near certainty (it was assumed that you were recently firing a rifle). Women were often subjected to rape.
Did they immediately replace the local government with something entirely new
Few officials would have stayed in office. There were a number of mayors that put local rightists in custody when the war began and kept violence to a minimum in their towns who were executed by the nationalists because their former prisoners claimed that they were about to be executed when the Nationalists arrived. Conditions were reverted to the arrangement prior to 1931, before the Republic began making reforms.
For your questions, the best source is Paul Preston's The Spanish Holocaust. He covers the 150,000 noncombatant deaths in Nationalist territory in detail, and the 50,000 noncombatant deaths in Republican territory in slightly less detail. He explains the ideological underpinnings of the actions taken by both sides as well as the horrifying consequences of carrying out those ideologies.
As always, followup questions from OP and others are always encouraged.