Understanding Mixer Operation
The Mixer datasheet from Analog devices does not talk about filtering
  the resultant signal post mixing


No it wouldn't because it assumes you are using it with a sine wave but harmonics will produce multiple images because a mixer is a signal multiplier.


  Or the Mixer by itself cuts out the higher frequency?


No, almost certainly not (some may do but I've never heard of one). You will need to bandpass either the output or use a sine wave for \$\omega_P\$.


  Also in Mixer stage, can I use clock signals instead of square wave as
  inputs?


You have to ensure that the DC content of the clock signal is either removed (capacitor in series) or that the mixer can handle the voltage range you propose.


  Is it possible to inject sine wave at the LO port. The datasheet is
  not explicit on this.


Almost certainly, this is the norm.