Could a larger capacitor, at the output of the SMPS, destroy components with larger current draw when its being charged?
Would a 100uF electrolitic capacitor draw an extremly high current at
  startup, as all capacitors do when they are being charged up and
  therefore would it destroy the IC or the inductor or would it be just
  fine because it's only high current for a fraction of a second?


No the IC would protect itself and, in addition, the LT8364 has a programmable soft-start option: -




  SS: Soft-Start Pin. Connect a capacitor from this pin to GND copper
  (near FBX) to control the ramp rate of inductor current during
  converter start-up. SS pin charging current is 2μA. An internal 220Ω
  MOSFET discharges this pin during shutdown or fault conditions.


For the TPS62125, if you read the data sheet it tells you that there is internal current limiting circuit (max 900 mA) - see page 5 of the data sheet - this is to prevent the inductor going into saturation primarily but serves the purpose you appear to want.

There is also a 200 us output voltage ramp time to reach 1.8 volts and this means that dV/dt = 9 V / ms so, if the capacitor is (say) 100 uF, the current cannot be higher than 0.9 amps (coincidentally as per the current limit circuit).

Generally, you can assume most modern switching controllers will have features that protect themselves against load currents that are too high and these are no exception.


  Would it be more effective to use a smaller value ceramic capacitor?


Use what is recommended in the data sheets = usually means use a ceramic for modern devices.