Do we neglect Gravity to conserve momentum of colliding bodies? Why do two bodies collide? What are the factors to be considered for collision?
It is dependant on velocities of the colliding bodies if they do not have a sufficient velocity then they will perform a projectile motion and fall not colliding due to the presence of gravity.
We say that momentum is always conserved when mass is constant and when net external forces acting on objects colliding are zero.
When collisions happen in air do we neglect gravity to conserve momentum? Why do we do this? Why is effect of gravity not taken into consideration?
 A: The collision is usually assumed to take place over a small amount of time.  If that's true we can ignore the effect of the acceleration due to gravity, as it has no time to change the velocities - and conservation of momentum holds true for the two bodies.
If the collision is not quick, conservation of momentum still holds true, but you would have to include the momentum of the earth.
A: The change of momentum due to an applied force is given by the impulse of the interaction.  This is true for any object in any interaction:
$$J = F\Delta t$$   During a typical collision problem, both $F$ and $\Delta t$ are small compared to the force of collision and the duration of the collision.  Hence, the impulse due to gravity can be neglected.
A: We approximate that a collision happens instantly, so there is no time for gravity (or anything else besides the collision) to change the momentum of either of the bodies.
Even if the collision is not instant (e.g. if you collide two Beanie Babies) gravity still can't have any effect since it acts on every part of each body the same way. They have no way to "detect" the force of gravity - until they hit the ground. All force detectors have to compare the acceleration of two different parts of the detector, and gravity affects both parts the same. This is actually the basis for Einstein's theory of relativity.
A: You are right. Momentum is not conserved along a direction where an external force is applied. Though the impulsive forces that occur at a collision are huge in comparison with the force due to gravity (weight). So the weight is negligible, therefore you can calculate velocities using conservation of momentum. And there is no matter when the collision happens horizontally, because at this time force due to gravity is perpendicular to the collision. So you can easily use the theory of conservation of momentum.
