A fully discharged capacitor, having a terminal voltage of zero, will initially act as a short-circuit when attached to a source of voltage, drawing maximum current as it begins to build a charge.
No. A capacitor does not EVER act as a short circuit when first connected. Anyone who tells you this is misinformed, or a poor teacher. "ICE" = Current leads Voltage across a capacitor. What this means is that electrons on either side of the capacitor move. On the positive side, they move away from the plate on that side, towards the power supply.
Why do capacitors get charged?
Most times the capacitors are not just put onto a full voltage but get charged as the supply voltage rises after power is applied. Some mains switch mode supplies have negative temperature coefficient surge limiting resistors in line with the mains for slowing down the inrush current. So it depends on the size of the capacitor and the application.
So, whenever you excite a discharged capacitor with a signal that "starts up", i.e. is zero before a given instant of time, the capacitor is subjected to a transient response that dies off after some time (depending on the time constant of the circuit).
The vertical wire drawn next to the vertical capacitor shorts the two terminals of the capacitor. Any current flowing through this circuit segment will flow through the vertical wire and completely bypass the vertical capacitor due to the short. This means you can ignore the shorted capacitor -- it has no effect on the circuit.
By having their shorted terminals, the voltage thereof is zero (more precisely, the potential difference between them), so that this element is not operational in the circuit, and can be removed for analysis. The other two capacitors are in series, hence that:
If at a given time t0 t 0 a capacitor is not charged, by definition, it has q(t0) = 0 q (t 0) = 0, hence v(t0) v (t 0) must be 0, even if some current in that instant of time is flowing. Therefore, at that instant, the capacitor is like a short circuit: current flowing, no resistance and no voltage across it.