I am trying to understand how upper and lower indices are connected when computing the energy-momentum tensor. In particular, I found the simple problem where the Lagrangian density is given as $$\mathscr{L}= \frac{1}{2}\eta_{\mu\nu}(\partial^{\mu}\phi)(\partial^{\nu}\phi)-\frac{1}{2}m^2\phi^2$$ and I want to compute the energy-momentum tensor $T^{\mu\nu}$. Beginning with the relation $$ T^{\mu\nu} = \frac{\partial \mathscr{L}}{\partial(\partial_{\mu}\phi)}\partial^{\nu}\phi-\eta^{\mu\nu}\mathscr{L}$$ one can find ${T_{\mu}}^{\nu} = \eta_{\mu\nu} T^{\mu\nu}$ and then afterwards go back to $T^{\mu\nu}$ through $T^{\mu\nu} = \eta^{\nu\mu} {T_{\mu}}^{\nu}$.
My questions are the following:
Is there a difference between $\frac{\partial}{\partial(\partial_{\mu}\phi)}$ and $\frac{\partial}{\partial(\partial^{\mu}\phi)}$, and if so how does one compute $\frac{\partial}{\partial(\partial_{\mu}\phi)} \frac{1}{2}\eta_{\mu\nu}(\partial^{\mu}\phi)(\partial^{\nu}\phi)$?
What happens to the term $\frac{\partial \mathscr{L}}{\partial(\partial_{\mu}\phi)}\partial^{\nu}\phi$ when multiplying by $\eta_{\mu\nu}$? Should any indices be moved? My current understanding is that I first need to find an expression for this not involiving $\frac{\partial}{\partial(\partial_{\mu}\phi)}$ and then just move all upper indices $\mu$ down.
As you probably can guess I am very new to this type of notation, and I would very much appreciate any help.
