Confused by raising and lowering indices I lack in understanding of some basic idea regarding 4-vectors and index raising and lowering. From what I understand that:
$$ \partial^\mu = \eta^{\mu\nu}\partial_\mu$$
So then, is the following correct?
$$ \partial_\mu\psi\partial^\mu\psi^* - \partial_\mu\psi^*\partial^\mu\psi =\partial_\mu\psi\partial^\mu\psi^* - \eta^{\mu\mu}\partial^\mu\psi^*\eta_{\mu\mu}\partial_\mu\psi  = \partial_\mu\psi\partial^\mu\psi^* - \partial_\mu\psi\partial^\mu\psi^*  = 0$$
EDIT:
The line above is apparently wrong. What I think I meant was:
$$ \partial_\mu\psi\partial^\mu\psi^* - \partial_\mu\psi^*\partial^\mu\psi =\partial_\mu\psi\partial^\mu\psi^* - \eta_{\mu\lambda}\partial^\mu\psi^*\eta^{\mu\lambda}\partial_\mu\psi  = \partial_\mu\psi\partial^\mu\psi^* - \partial_\mu\psi\partial^\mu\psi^*  = 0$$
where $*$ is just a complex conjugate. 
And if this is correct, is this then true:
$$\eta^{\mu\lambda}\eta_{\mu\lambda} = \delta $$
 A: There are some index issues with your expressions as mentioned by others.  The correct way to raise/lower an index is by
$$ \partial^\mu = \eta^{\mu\nu}\partial_\nu .$$
Notice how the free index $\mu$ appears in the up position exactly once on each side of the equal sign, and the dummy index $\nu$ is repeated in the up and down position on the same side.
The dummy index $\nu$ is summed over, so the expression should be read as:
$$ \partial^\mu = \sum_{\nu=0}^3 \eta^{\mu\nu}\partial_\nu. $$
The expression tells you how to get each of the 4 components of $\partial^\mu$.
When we apply that to the relevant part of your second expression we get:
$$ \left(\partial_\mu \psi^*\right) \, \left( \partial^\mu \psi \right)
  = \left( \eta_{\mu\nu} \partial^\nu\psi^*\right) \, \left( \eta^{\mu\lambda} \partial_\lambda \psi \right).
$$
To get the indices correct we make sure that each term in parentheses maintains a single $\mu$ index in the correct position, and that in a given multiplicative statement no index is used more than once in a given up/down position.  These means we need a different dummy index for each summation, otherwise we might not know which terms get summed with which.
The final statement is then:
\begin{align}
\partial_\mu \psi \partial^\mu \psi^*
    - \partial_\mu \psi^* \partial^\mu \psi 
  &= \partial_\mu \psi \partial^\mu \psi^*
    - \eta_{\mu\nu} \partial^\nu\psi^* \eta^{\mu\lambda} \partial_\lambda \psi \\
  &= \partial_\mu \psi \partial^\mu \psi^*
    - \left( \eta_{\mu\nu} \eta^{\mu\lambda}  \right) \partial^\nu\psi^* \partial_\lambda \psi \\
  &= \partial_\mu \psi \partial^\mu \psi^* - \partial_\nu \psi \partial^\nu \psi^* \\
  &= \partial_\mu \psi \partial^\mu \psi^* - \partial_\mu \psi \partial^\mu \psi^* \\
 &= 0
\end{align}
We used the fact that
$$\eta_{\mu\nu} \eta^{\mu\lambda} = {\delta_\nu}^\lambda,$$
and that dummy indices can be renamed arbitrarily ($\nu\rightarrow\mu$).
Finally as a commenter stated, if you fully contract a metric with itself you get:
$$\eta_{\mu\nu} \eta^{\mu\nu} = {\delta_\nu}^\nu = 4$$
in a 4-D space.
A: No. You cannot have more than one of the same index raised/lowered position. If you want to apply the metric to second set of terms, the second line should read
\begin{equation}
....=\partial_\mu \psi \partial^\mu \psi^* - \eta_{\mu\nu} \partial^\nu \psi^* \eta^{\mu\lambda}\partial_\lambda\psi = ....
\end{equation}
