Can linear momentum be conserved before and after collision in the presence of an external force?

Consider your example 1 with two billiard balls, mass $0.16 \,\rm kg$, colliding with a red ball falling down with a speed of $5\, \rm m\,s^{-1}$ and colliding with a stationary white ball.
Applying conservation of linear momentum (assuming there are no external force and the collision is elastic) results in the red ball momentarily stopping and the white ball moving downwards at $5\, \rm m\,s^{-1}$.
The impulse (change of momentum) on each ball is $0.16 \times 5 = 0.75 \, \rm N\,s$.

During the collision the gravitational force would have had a effect on the balls but to know how much one must know the collision time.

Peter Bohacek has produced many Direct Measurement Videos and the relevant one for this answer is Billiard Ball Collision three consecutive frames from which are shown below.

enter image description here

This shows that the collision time is less than $0.001 \, \rm s$.

Going back to the falling red and white billiard balls, in a time of $0.001 \, \rm s$ the impulse due to gravity on one of the balls is $\text{~}\, 0.16 \times 10 \times 0.001 = 0.016 \,\rm Ns $ which is very much smaller than the impulse on the balls due to the collision, $ 0.75 \, \rm N\,s$.
So the assumption of a very short collision time resulting in very little effect on the outcome of the collision is a good one.

More time could have been spent analysing the video to get a more accurate upper bound for the collision time which might be the basis of a nice assignment?

You are basically telling them the right thing. In all three situation if we assume that gravitation is just an external force, momentum is not conserved. The reason is that we don't take the momentum transfer between the object in the system to the source of the external force (in this case the earth) into account.

If two bodies in our system interact for a very short time we can still assume momentum conservation for this interaction. The reason is that the momentum transfer between the objects in our system is much larger than the momentum transfer to the objects outside our system during this brief time period.