Gain vs. Directivity and effect of animal body to Antennas
Question 1: Does the gain of an antenna always increase with the
  decrease in directivity?


Real antenna gain is nearly always referred to the theoretical isotropic antenna. The isotropic antenna emits power in all directions equally therefore it projects power onto the surface of a sphere where the antenna is at the centre of the sphere.

At distance r (radius of sphere), the power from an isotropic antenna is passing thru a spherical area of \$4 \pi r^2\$ square metres.

Normal antennas (such as dipoles) do not transmit this power in all directions therefore they are said to have a gain in certain directions compared to the isotropic antenna and, indeed there is more power per sq metre at a comparable distance, but this is beginning to become "directional". Therefore the higher the directionality of an antenna, the more power it concentrates in one direction (reciprocal for receiving antennas too) and the higher the gain.


  Question 2 : What type of GSM antenna is recommended for an
  application that can run in such low GSM coverage areas?


What is the likely incident power received and what is the minimum power needed by the receiver. A good figure for required power by the receiver is based on the signal data rate: -

Received power is -154dBm + 10\$log_{10}\$(data rate) - from this you can calculate the headroom, add maybe 20 dB for fade margin (could be lower if you accept a longer delay and you are moving).


  Question 3 : What sort of an effect would the animals giant body have
  in the reception of GSM signals? If yes, what are the remedies I can
  do to reduce the impact of the same.


I have no idea.