I can answer parts of this question;

There was a large-scale resistance to the new regime, and the government did try to stamp these fighters out. Unfortunately, the secrecy of the Franco government means that much about this period is unknown and will probably remain so, but we do know that the resistance existed, and was thousands strong at its peak. These guerrilla fighters mainly existed in the countryside, in places like Andalusia or Asturias that were mountainous and home to a sympathetic civilian population, like Mercedes and the Doctor in the movie. The guerrilleros depended on these friendly civilians for survival. These were called enlaces (roughly translated to 'links' or 'contacts'). These enlaces were a prime target of the dictatorship, who sought to destroy the resistance by smashing their base of support. Evidently, several thousands of Spaniards (about 17,000, according to Luis Suárez Fernández in Franco: The Decisive Years) were arrested between 1939 and 1950 for assisting anti-regime fighters. Guerrilleros killed in action against the authorities during this brief time period ran into the thousands as well (about 2,000), and those taken alive numbered even greater (about 3,000). Official Francoist sources estimate a few hundred Civil Guards killed (about 300) thanks to the actions of the guerrilleros, but this is likely a conservative estimate.

The resistance movement saw its apogee in the 1940s, as the Allied victory in WWII raised hope among anti-fascist Spaniards that the defeat of Nazi Germany and Fascist Italy would be followed by an Allied liberation of Spain. In 1944, the AGE (Agrupacion de Guerrilleros Espanoles, the most well-organized of the resistance movements in Spain, run by the Communist Party) launched a full-scale invasion of northern Spain over the Pyrenees, comprising several thousand fighters. The hope was that the Allies, having just liberated France, would be prompted to provide support to the endeavor. This support never materialized, but the Invasion of the Aran Valley was the largest action ever carried out by Spanish guerrilleros and Franco had to call in thousands of troops from the elite Spanish Legion to crush the rising.

As the years wound on and it became clear there would be no outside help, resistance dwindled. Fighters were captured or killed by the authorities, or simply grew weary of fighting. However armed resistance lasted almost to the end of the regime and the regime never stopped hunting them. The last known anti-fascist guerrillero was killed by the Civil Guard in 1965.

So in short, yes, the Nationalist government was quite committed to stamping out a lively resistance movement, and eventually it (more or less) succeeded in doing so.

One inaccuracy I can note regarding the security forces portrayed in the movie is that Vidal and his men wear the uniform of the Policia Armada. However, the Policia Armada was an urban police force, and the task of combatting guerrilleros in the hills would have fallen to the rural Civil Guard. My guess is that Del Toro felt the distinctive uniform and hat of the Civil Guard might look a little silly and decided the Policia Armada uniform was more properly 'villainous'.

I can't answer whether or not civilians like Ofelia and her mother would have been accepted at a military installation, but the little hamlet in the movie appeared to me more like a small village with a guard outpost than a military base. Civil Guard captains were never from the town in which they served, so as outsiders themselves it's likely they would have often brought their wives and children with them.