Picture below is from Topping's Lectures on the Ricci flow.
First, I don't know why $$ (\pm*d*\alpha) dV = \pm d(*\alpha), $$ in fact, I don't know what is $*\alpha$ and $d*\alpha$. In my opinion, $*$ should be linear combination of tensor times with contraction and switch the type. Therefore, in general, we use denotation as $1*A, A*B$ where $A,B$ are tensor. But in $d*\alpha$, $d$ is an operator. Besides, $*\alpha$ is harder to understand.
Second, since $\delta(T)=- tr _{12} (\nabla T)$, I have $$ \delta(f\alpha) = -g^{ij} \nabla_{\partial_i} (f\alpha)(\partial_j) = -g^{ij}\partial _i [f\alpha(\partial_j)] +g^{ij}f\alpha(\nabla_{\partial_i}\partial_j) = f(\delta \alpha) - g^{ij} \alpha(\partial_j) df(\partial_i) = f(\delta \alpha) - \langle df, \alpha\rangle $$ where $\alpha$ is 1-form. I always use to calculate in a local coordinate, seeemly, it makes thing becomes complex. How should I practice my calculation better ?
