One of the assumptions of General Relativity is a vanishing torsion tensor. One can, for example, modify the formalism to accommodate torsion by introducing a "torsion potential" to the vielbein, or equivalently gauging the translation group.

Nonzero torsion in physics is seen in teleparallel theories of gravity on Weitzenböck spacetime and in Einstein-Cartan Theory which generalizes GR. I'm not sure what you mean by "Geometrical Picture," but a parallel transport over a closed path in a space with torsion produces a displacement from the original position, like how curvature changes a vector's orientation. On a discrete lattice, this means that torsion is equivalent to dislocation defects.

The Christoffel symbols can't be derived purely from the metric tensor in this case. See https://en.wikipedia.org/wiki/Contorsion_tensor