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Do Dirac Gamma Matrices act like Tensors? Is it true that
$$ \gamma_\mu = \eta_{\mu\nu}\gamma^\nu~? $$
Also what about $\sigma_{\mu\nu}$, where $\sigma_{\mu\nu}$ is defined to be: \begin{align*} \sigma_{\mu\nu} = \frac{i}{2}[\gamma_\mu,\gamma_\nu]~? \end{align*}
Yes. The indices on gamma matrices can be treated like four-vector indices.
In particular, indices on gamma matrices are commonly raised and lowered with the Minkowski metric $\eta_{\mu\nu}$ as you indicate; \begin{align} \gamma_\mu = \eta_{\mu\nu}\gamma^\nu. \end{align} Now, as user26143 writes in his comment above, the gamma matrices have the following interesting property: \begin{align} \Lambda_{\frac{1}{2}}^{-1}\gamma^\mu\Lambda_{\frac{1}2} = \Lambda^\mu_{\phantom\mu\nu}\gamma^\nu \end{align} where $\Lambda^\mu_{\phantom\mu\nu}$ are the components of a Lorentz transformation in the defining (four vector) representation of the Lorentz group, and $\Lambda_\frac{1}{2}$ is the matrix representation of this Lorentz transformation in the Dirac spinor representation. This equation simply says that when one transforms the Dirac spinor indices of the gamma matrices (these indices are suppressed on the left-hand side of the above equation) then this is equivalent to a transformation of its vector index. This fact does not somehow invalidate treating the Greek indices that label the gamma matrices as Lorentz four-vector indices.
It follows from this that indices on composite objects formed by the gamma matrices should also be considered Lorentz vector indices. This is, in particular, true for the object $\sigma^{\mu\nu}$ you define above whose indices can therefore also be lowered using the Minkowski metric.
The expressions $\eta_{\mu\nu}$ are some numbers. The second relation you posted already implies that people multiply and add different gamma matrices, and mutliply gamma matrices by complex numbers, and for any fixed $\mu$ the expression the second expression is just $\eta_{\mu 0}\gamma^0+\eta_{\mu 1}\gamma^1+\eta_{\mu 2}\gamma^2+\eta_{\mu 3}\gamma^3$. So your question isn't really about the "behaviour" of Gamma matrices, it's merely about if the academic community needs introduction to your notation.
