Qualitatively it is very simple.
In your top figure, The capacitor starts charging at a rate determined by the time constant. Because the R is rather small, the voltage across the capacitor is following the sine curve and when it reaches the breakdown voltage, the spark takes place and acts like a short.
As soon as the spark takes place, it becomes a dead short and we have the LC circuit complete. As the capacitor is already charged, high frequency oscillations will take place (depending on the LC time constant) and this will induce very high voltage on the secondary of the tesla coil. This will be a RF frequency (at the secondary of the tesla coil).
About the same considerations for the second figure: The LC cannot discharge via the secondary of the power transformer because it is acting as the voltage source. However, as soon as the voltage reaches the maximum for the breakdown of the spark gap, again the same LC oscillations will take place. These two circuits are identical for all practical purposes.
About the question: why the sparking is not continuous? Because the voltage is sinusoidal and as soon as the voltage comes close to the value (becomes less than) required for the spark to sustain, the spark is extinguished.
A common neon indicator lamp also flickers are the main frequency (100-120 Hz)- and so are the common fluorescent lamps (not the CFL- they have an internal HF oscillator). This is a feature of all discharge lamps that run on AC. When we use high frequency AC, the voltage comes up before the discharge has time to extinguish- they appear continuous.