When aren't Christoffel symbols symmetrical with respect to bottom indices, and why? When aren't Christoffel symbols symmetrical with respect to the bottom indices? Why isn't the symmetry of second derivatives true in this case?
 A: Let $\pi:E\rightarrow M$ be an arbitrary vector bundle, not necessarily the tangent bundle $TM$ of $M$. Locally, say over an open subset $U\subset M$, one can choose a family of point-wise linearly independent sections $\{E_i:U\rightarrow E\}$, which is called a local frame in $E$ (over $U$). Then each section $X:U\rightarrow E$ can be uniquely represented in terms of $\{E_i\}$
$$
X=X^i E_i \quad \mbox{(Einstein summation assumed)}
$$
In presence of a connection $\nabla$ in $E$ the covariant derivatives can be computed in terms of the Christoffel symbols of $\nabla$ with respect to the local frame $\{E_i\}$ defined as
$$
\nabla_{E_i}{E_j} = \Gamma^k_{ij} E_k
$$
In general, the Christoffel symbols need not to be symmetric since torsion may not vanish, as noted by the commenters.
For the case of Riemannian manifolds, the Riemannian metric $g$ induces a preferred connection in the tangent bundle which is called the Levi-Civita connection. We prefer this one because it is torsion-free (the torsion tensor $T(X,Y)=\nabla_X Y - \nabla_Y X -[X,Y]$ vanishes) and the metric tensor is parallel w.r.t. to it: $\nabla g =0$.
Connections in the tangent bundle are sometimes termed linear.
In a coordinate patch $(U,x^i)$ we have the standard coordinate frame $\{ \partial_i \}$ (of the tangent bundle) where vector fields $\partial_i$ act as partial derivatives on functions. This implies that a linear connection is torsion-free if and only if the Christoffel symbols with respect to any coordinate frame (!) are symmetric  w.r.t the bottom indices (see J.Lee, Riemannian manifolds. An introduction to curvature, Springer 1987, p.63) (Hint: $[\partial_i, \partial_j]=0)$
