order of tensor indices vs their variance properties (upper vs lower) If I write a tensor (in a context where there's some metric $g_{ij}$) as, ${T^i}_j$, this 'means' that the $i$ components are contravariant, and the $j$ components are covariant with respect to the basis vectors ($e_k$).  Contravariant components correspond to 'column' vectors and covariant components to 'row' vectors, so we might also write,
$$\vec{v} = a^i e_i = \pmatrix{e_0, \cdots, e_n } \pmatrix{a_0\\\cdots\\a_n}.$$
This seems to imply that in ${T^i}_j$, $i$ corresponds to the rows (because it's a lower-index) and $j$ corresponds to the columns (upper-index).  This seems contrary to another convention (or fact?) that the first index refers to the rows and the second index to the columns.  If the latter convention is true (order reflects orientation), then this seems to imply that $({T^i}_j)^T = {T_j}^i$.  However, if the former convention is true, it seems to imply that $({T^i}_j)^T = {T^j}_i$.  But these are only equivalent in the special case of a Minkowskian metric (i.e. $g_{ij} = \delta_{ij}$).
Are these incompatible conventions?  If we are treating upper indices as  contravariant (columns), and lower indices as covariant (rows), then does the order of indices not matter at all?
 A: As far as I can tell, two different kinds of notation are being conflated here.
In index notation, there is no need notion of "row" or "column". An expression like $v^i=T^{i}{}_{jkl}a^jb^kc^l$ makes sense without specifying any particular way of arranging all of the components into rows and columns (and in that case there is no obvious way to do so). We still have covariant and cotrovariant indices, which simply specify how the elements transform when we change basis, but these aren't entirely the same as row/column as they are used in matrix notation.
Expressions from matrix notation can by translated in a standard way to index notation, and in these cases the notion of a "row index" and "column index" makes sense, but there are many tensorial expressions which cannot be written in terms of rows and columns.
Additionally, the matrix transpose isn't really used in the same way in tensor notation, more or less for the reasons that you state: simply "flipping" the elements of a $(1,1)$ tensor is not independent of the choice of coordinates. With an inner product (metric), we can do something similar by reversing the indices and raising and lowering accordingly, i.e.
$$
\left(T^T\right)^i{}_j=g^{ik}g_{jl}T^l{}_k
$$
but this won't be the same as the matrix transpose except in orthonormal coordinates
