Will this short range AM transmitter constructed using crystal work?
Now my question is whether this circuit will be able to generate the
  carrier frequency to transmit the audio input. If not, then what are
  all the required additions and corrections?


My advice is do some research and forget your circuit. Google is your friend. Type "crystal AM modulator" and look at the images that it finds: -



Next choose one that appears to give good information about how it works and try it out. Also be very aware of your country's laws on illegal transmitters.


  Also can anyone explain how to design the receiver to receive the
  signals.


Get your transmitter working on a band that a regular transistor radio works on then think about designing a receiver (much trickier).