I'm confused about where we should put tensor indices when we vary an action wrt the metric. For example, if I have in the Lagrangian a term such as $$ A_{\mu\nu}B^{\mu\nu}, $$ do I necessarily have to write it as $g^{\rho\mu}g^{\nu\sigma}A_{\mu\nu}B_{\rho\sigma}$ before performing the variation? My guess is that if (prior to the definition of the Lagrangian) A is defined as a (0,2) tensor and B as a (2,0) tensor, then $A_{\mu\nu}B^{\mu\nu}$ doesn't depend on the metric, whereas things like $$ F_{\mu\nu}F^{\mu\nu} = g^{\rho\mu}g^{\nu\sigma}F_{\mu\nu}F_{\rho\sigma} $$ do depend on the metric as one needs to map $F_{\mu\nu}$ to its dual space to be able to contract it with itself.
Is the above correct?
What if the quantities being contracted aren't tensors (thus cannot be mapped back and forth from/to tangent/cotangent space), eg terms formed by contracting the Christoffel symbol $\Gamma^\alpha_{\mu\nu}$ with itself? I'd assume there wouldn't be any metric dependence of the form $\Gamma^\alpha_{\mu\nu} = g^{\alpha\beta}\Gamma_{\beta\mu\nu}$.