If I'm right ,we define the inner product of tensor on Riemannian manifold as follow: $$ <T_{ij}^k,T_{ij}^k>=g_{il}g^{jm}g^{kp}T_{jk}^iT^l_{mp} $$ So ,if the metric $g_{ij}$ evolve under Ricci flow $g_{ij}=-2R_{ij}$, when I do $\partial_t <R_{ij},R_{kl}>$,it should be $\partial_t (g^{ik}g^{jl}R_{ij}R_{kl})$.
But it seemly be $\partial_t <R_{ij},R_{kl}>=<\partial_t R_{ij},R_{kl}>+ <R_{ij},\partial_t R_{kl}>$ in the below paper's 191 page.
When I computer the inequality 1 and 2, it seemly imply $\partial_t<Rm,Rm>=2<\partial_t Rm,Rm>$.
Why the $\partial_t$ and inner product can be exchanged ?
This is not true - you definitely need to take the variation of the metric in to account, as you correctly observe in your first few lines. You shouldn't need it to be true to get these inequalities - the extra terms in $\partial_t |A|^2$ due to the variation of the metric will be of the form $Rc * A * A$ and thus can be controlled by the $C |Rm||A|^2$ term.