Voltage Drop due to inrush current
Why does this voltage-drop occur? Does it have to do with the long transmission lines? 








simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. A simple model of the generation, transmission and utilisation of an electrical supply.

Yes. The transmission line has some resistance (R1) as well as some inductance (L1). Both of these will cause an increasing voltage drop along the line in proportion to the current drawn.

When R2 is switched on it forms a potential divider and, ignoring inductance for simplicity, the voltage at the house will be given by
$$ V_h = \frac {R_2}{R_1+R_2}V_1 $$
To maintain a steady voltage at the house R1 needs to be kept low relative to the maximum load (minimum resistance) of the load in the house.

The situation is aggravated by sharing the supply with several other buildings on your street. 


  Does this voltage drop happen even when a normal electrical appliance (like another bulb) is switched on?


Yes, but for small loads (high R2) R1 becomes proportionally less significant and the voltage drop won't be noticeable.


  When an air-conditioner is switched on, light bulbs in the house flicker a little.


The main component of your air-conditioning unit is the motor. Motors draw increasing current with increasing load. That means that as you slow them down the current increases. The worst moment is on switch-on when the motor draws up to ten times the current it draws while running. This momentary high load is what causes the voltage to sag.