No.

Consider the example of a right circular cylinder as the surface in question and let $C$ be a "generator" (i.e., a line on the cylinder parallel to the axis). Then the geodesic curvature is everywhere zero, but if epsilon is greater than half the circumference there is nor epsilon-tubular neighborhood.

No.

Consider the example of a right circular cylinder as the surface in question and let $C$ be a "generator" (i.e., a line on the cylinder parallel to the axis). Then the geodesic curvature is everywhere zero, but if epsilon is greater than half the circumference there is no epsilon-tubular neighborhood.

What if the surface is a right circular cylinder of very small radius and the curve is a generator?

No.

Consider the example of a right circular cylinder as the surface in question and let $C$ be a "generator" (i.e., a line on the cylinder parallel to the axis). Then the geodesic curvature is everywhere zero, but if epsilon is greater than half the circumference there is nor epsilon-tubular neighborhood.