Different antenna lengths for different frequencies?
Does FM and AM require different antenna sizes?


Not only will they be different lengths but usually completely different types of antennas. For instance, a medium/long wave (circa 100 kHz to a few MHz) receiving radio will use a ferrite rod antenna like this: -



And this antenna is totally unsuitable for FM transmissions in the 88 MHz to 108 MHz band (and above). FM radio will use a monopole like this: -



Different types of FM radios may use a dipole like this: -



Or even a folded dipole like this: -



You can even use Yagi Uda antennas like this: -




  How does a single FM antenna length works in the entire FM frequency band range, when there are theories which state that the antenna size should always be close to wavelength/4 for proper reception &amp; transmission?


It's a compromise - it will tend to be perfectly "on-song" at 98 MHz and slightly less "on-song" at 88 MHz and 108 MHz. The signal it pulls from the air will be about the same amplitude at either end of that spectral range but the impedance characteristic of the antenna will vary across the range. Good receiver design ensures this won't be a problem: -




  Does the antenna length scale itself within that range or how does it
  work?


No, it's a compromise.


  Why does low frequency signal have high range but high frequency does
  not?


The transmitted range is exactly the same but, because a receiving quarter wave antenna (for example) will be proportionally smaller at higher frequencies, the "net" that captures radio energy is also proportionally smaller and hence it captures less emitted power. Think about an LED and a photosensitive detector. If the detector only has a small aperture to let light in, it will pick up less light power than a detector that has a large surface area.