Are neutrinos affected by gravity?
If not, could that be a plausible reason for a neutrino taking a shorter path than light, since light is affected by gravity?
Everything is affected by gravity. Gravity warps of space-time according to the Einstein Field Equations, and traveling on "geodesics" (shortest path curves) on that curved surface is how gravity is manifested.
Thinking as though there is some sort of euclidean space underneath the non-euclidean space-time in which neutrinos can take a more direct "straight line" between the points is not at all supported. Everything we know of travels on this curved spacetime.
Also, speaking more Newtonianly, neutrinos do seem to have mass in the more classic sense.