Transistors - Small signal and Large signal analysis
Here are my answers to your questions (1)...(5): 

(1) Capacitors can be considered as short circuits if their impedance is much smaller than the resulting total (effective) ohmic resistance which appears in series or in parallel to the capacitor. The meaning of "much smaller" depends on the allowed calculation error (and the associated parts tolerances).

(2) Small-signal analyses are used for finding the gain and the input-/output impedances of a circuit. These 3 parameters are signal RATIOS (voltage/voltage current/current or voltage/current), which can be found only if these input/output quantities have the same sinusoidal waveform. Otherwise, we cannot calculate a ratio. And this is ensured for small-signals only (linearized V-I characteristics).

(3) Large signal analysis is required if the signals are so large that the preconditions for small-signal analyses (linearity) are not met anymore. However, in this case, no gain and no input/output impedances can be calculated. However, we can define and calculate input-/output power and efficiency values. (Examples: Transistorized power stages or switches).

(4) We have no choice. See (3).

(5) The design process depends on the particular circuit and its purpose (linear gain stage, power stage, switch,...)