Coordinate conditions in black hole solutions
To start with, given the metric ansatz $$ ds^2 = e^{2\gamma(u)}dt^2 - e^{2\alpha(u)}du^2 - e^{2\beta(u)}d\Omega^2,\tag{1} $$ one can apply the Cartan-Karlhede algorithm to check local equivalence with the Schwarzhschild solution (this is best carried out with a computer implementation). Instead of this general solution for the local equivalence of some metric ansatz and a given metric, one can in this case also investigate the demands for (1) to be a vacuum solution, since this will make it the Schwarzschild solution by virtue of being static and spherically symmetric (Birkhoff's theorem); the latter approach should of course also be used if one is primarily interested in the requirements for an ansatz to give a solution rather than some specific soluion.
To prove that $f(\alpha,\beta,\gamma) = 0$ is a "gauge" you can plug it into your ansatz and take the approach outlined above. For example for the ansatz (1) any "gauge" that does not violate the equations below is allowed.
I am new to gauge theory, but I believe the correct use of gauge transformations in general relativity would entail $SO_0(1,3)$-transformations of vectors in a rigid frame. Thus I believe you are correct in that this is a matter of coordinate freedom rather than gauge freedom.
Here I approach the metric ansatz (1) as an example: to that end I will be working in a local Lorentz (orthonormal) frame: \begin{align}\begin{split} \omega^0 &= e^{\gamma(u)}dt, \\ \omega^1 &= e^{\alpha(u)}du, \\ \omega^2 &= e^{\beta(u)}d\vartheta, \\ \omega^3 &= e^{\beta(u)}\sin(\vartheta) d\varphi. \end{split}\tag{2} \end{align} We then work with the first Cartan equation $$ d\omega^i = \gamma^i{}_{jk}\omega^j \wedge \omega^k,\tag{3} $$ where $\gamma^i{}_{jk}$ are the Ricci rotation coefficients (components of the connection forms). Applying (3) to (2) we find \begin{align} \gamma^0{}_{jk}\omega^j\wedge\omega^k &= \gamma'(u)e^{\gamma(u)}du \wedge dt \\ &= \gamma'(u)e^{-\alpha(u)}\omega^1 \wedge \omega^0, \\ \gamma^1{}_{jk}\omega^j\wedge\omega^k &= 0, \\ \gamma^2{}_{jk}\omega^j\wedge\omega^k &= \beta'(u)e^{\beta(u)}du\wedge d\vartheta \\ &= \beta'(u)e^{-\alpha(u)}\omega^1 \wedge \omega^2, \\ \gamma^3{}_{jk}\omega^j\wedge\omega^k &= \beta'(u)e^{\beta(u)}\sin(\vartheta)du\wedge d\varphi + e^{\beta(u)}\cos(\vartheta)d\vartheta\wedge d\varphi \\ &= \beta'(u)e^{-\alpha(u)}\omega^1 \wedge \omega^3 + \cot(\vartheta)e^{-\beta(u)}\omega^2 \wedge \omega^3, \end{align} and by inspection we find \begin{align} \begin{split} \gamma^0{}_{10} &= \gamma'(u)e^{-\alpha(u)}, \\ \gamma^2{}_{12} &= \beta'(u)e^{-\alpha(u)}, \\ \gamma^3{}_{13} &= \beta'(u)e^{-\alpha(u)}, \\ \gamma^3{}_{23} &= \cot(\vartheta)e^{-\beta(u)}, \end{split}\tag{4}\end{align} to be the only non-zero coefficients (up to symmetries: $\gamma_{ijk} = \gamma_{[ij]k}$). Next the second Cartan equation \begin{align} \frac{1}{2}R^i{}_{jk\ell}\omega^k \wedge \omega^\ell &= d\gamma^i{}_j + \gamma^i{}_k \wedge \gamma^k{}_j,\tag{5} \\ &= \left(-\gamma^i{}_{j[k|\ell]} + \gamma^i{}_{jm}\gamma^m{}_{[k\ell]} - \gamma_{mj[k}\gamma^{im}{}_{\ell]}\right)\omega^k \wedge \omega^\ell \end{align} gives, using (4): \begin{align}\begin{split} R^0{}_{101} &= \left(\alpha'(u)\gamma'(u) - \gamma''(u) - \gamma'(u)^2\right)e^{-2\alpha(u)}, \\ R^0{}_{202} &= -\beta'(u)\gamma'(u)e^{-2\alpha(u)}, \\ R^0{}_{303} &= -\beta'(u)\gamma'(u)e^{-2\alpha(u)}, \\ R^1{}_{212} &= \left(\alpha'(u)\beta'(u) - \beta''(u) - \beta'(u)^2\right)e^{-2\alpha(u)}, \\ R^1{}_{313} &= \left(\alpha'(u)\beta'(u) - \beta''(u) - \beta'(u)^2\right)e^{-2\alpha(u)}, \\ R^2{}_{323} &= e^{-2\beta(u)} - \beta'(u)^2e^{-2\alpha(u)}, \end{split}\tag{6}\end{align} as the only non-zero Riemann components, again up to symmetries ($R_{ijk\ell} = R_{([ij][k\ell])}$), or in terms of the Ricci components (suppressing function arguments): \begin{align}\begin{split} R_{00} &= \left(2\beta'\gamma' - \alpha'\gamma' + \gamma'' + \gamma'^2 \right)e^{-2\alpha}, \\ R_{11} &= \left(\alpha'\gamma' + 2\alpha'\beta' - \gamma'' - 2\beta'' - \gamma'^2 - 2\beta'^2 \right)e^{-2\alpha}, \\ R_{22} &= \left(\alpha'\beta' - \beta' \gamma' - \beta'' - 2\beta'^2 \right)e^{-2\alpha} + e^{-2\beta}, \\ R_{33} &= \left(\alpha'\beta' - \beta' \gamma' - \beta'' - 2\beta'^2 \right)e^{-2\alpha} + e^{-2\beta}, \\ \end{split}\tag{7}\end{align} The Ricci tensor must vanish in a vacuum solution (assuming no cosmological constant for simplicity), so (7) gives us a system of three differential equations: \begin{align} 2\beta'\gamma' - \alpha'\gamma' + \gamma'^2 + \gamma'' &= 0, \tag{A}\\ \alpha'\gamma' + 2\alpha'\beta' - \gamma'' - 2\beta'' - \gamma'^2 - 2\beta'^2 &= 0, \tag{B}\\ \alpha'\beta' - \beta' \gamma' - \beta'' - 2\beta'^2 &= -e^{2(\alpha - \beta)}\tag{C}. \end{align} Notice that equations (A), (B), and (C) can be simplified:
If $\gamma' = 0$ we end up with a single equation: $$ e^{\alpha} = \beta'e^\beta, $$ under which all Riemann components (6) vanish. So this corresponds to flat (Minkowski) space.
Otherwise, by solving for $\gamma''$ in (A) and (B) we find $-\beta'\gamma' = \alpha'\beta' - \beta'' - \beta'^2$, allowing us to write \begin{align} \frac{\beta''}{\beta'} &= \alpha' - \beta' + \gamma', \tag{A*}\\ \frac{\gamma''}{\gamma'} &= \alpha' - 2\beta' - \gamma', \tag{B*}\\ 2\beta'\gamma' + \beta'^2 &= e^{2(\alpha - \beta)},\tag{C*} \end{align} because $\beta' \neq 0$ by necessity, and we can integrate (A*) and (B*) to find \begin{align}\begin{split} \beta' &= C_1e^{\alpha - \beta + \gamma}, \\ \gamma' &= C_2e^{\alpha - 2\beta - \gamma}, \end{split}\tag{D}\end{align} under which (C*) becomes $$ C_1e^{-\alpha + \beta + \gamma}\left(2C_2e^{\alpha - 2\beta - \gamma} + C_1e^{\alpha - \beta + \gamma}\right) = 1, $$ or $$ e^{2\gamma} = C_1^{-2}\left(1 - 2C_1C_2e^{-\beta}\right)\tag{E1}. $$ Note that (D) has already given us: $$ e^{2\alpha} = C_1^{-2}\beta'^2e^{2(\beta - \gamma)}\tag{E2}. $$
Equations (E1) and (E2) allows us to write (1) as $$ ds^2 = \left(1 - 2Me^{-\beta(u)}\right)dt^2 - e^{2\beta(u)}\left(\frac{\beta'(u)^2}{1 - 2Me^{-\beta(u)}}du^2 + d\Omega^2\right),\tag{1.1} $$ both by rescaling coordinates by constant factors if necessary. Notice that we have performed no (non-trivial) coordinate change. Thus (1.1) is just (1) rewritten to conform with the EFEs, excluding flat space, with an aptly chosen name on the integration constant.
We observe that imposing $\alpha(u) = -\gamma(u)$ is equivalent to restricting to $\beta' = e^{-\beta}$ or equivalently $\beta = \log(u + C)$ or $e^{2\beta} = (Cu)^{2}$ (new $C$), by (E2), which is just the Schwarzschild coordinates for (1.1). Similarly, from (B*) we can see that imposing $\alpha(u) = 2\beta(u) + \gamma(u)$ is equivalent to imposing $\gamma = Au + B$, which allows us to rewite (1.1) as (barring any calculation errors) $$ ds^2 = e^{2Au}dt^2 - \frac{1}{\left(1 - C^2e^{2Au}\right)^{2}} \left(\frac{e^{2Au}}{\left(1 - C^2e^{2Au}\right)^2}du^2 + d\Omega^2\right). $$ For example $\alpha(u) = -\gamma(u) + \beta(u)^2$ can similarly be plugged into e.g. (E2) to yield $$ \int C_1^2 du = \int e^{2\beta - \beta^2} d\beta . $$
As another example the condition $\beta = 2\gamma$ can be seen to demand a constant $\beta$, from (E1) and so cannot be used.
To be more general, any condition on the form $\alpha = f(\beta,\gamma)$ gives a first order ODE for $\beta$, by (E1) and (E2). By the standard existence theorem such a condition is consistent with the EFEs. On the other hand a condition on the form $\beta = f(\gamma)$ gives an equation for $\beta$, and so is not consistent with the EFEs. The consistency of imposing more than one condition can also be checked with the above equations. For example $\alpha = \beta = \gamma$ can immediately be seen to not be consistent with the EFEs, by (D).
What Erik has said is correct, essentially you write down an ODE (for your case) but PDE in general and see if there are solutions for which your 'gauge' choice holds. This is exactly a coordinate-freedom, perhaps this is where the confusion for question 2 comes in.
In GR there is a second kind of gauge freedom known as tetrad invariance, which is invariance under reference frame transformations. Essentailly you want the metric tensor to be invariant under a tetrad transformation. For example, in the Newman-Penrose formalism we write the metric in terms of the null tetrad $(\boldsymbol{l},\boldsymbol{n},\boldsymbol{m},\boldsymbol{\bar{m}})$ \begin{equation} g_{ab}=-l_a n_b - n_a l_b +m_a \bar{m}_b +\bar{m}_a m_b, \end{equation} which we want to be invariant under a transformation of the basis (e.g. $m \rightarrow e^{i \vartheta} m$ and $\bar{m} \rightarrow e^{-i \vartheta} \bar{m}$ for some angle $\vartheta$). Note the absence of coordinates. The term `gauge invariant' is usually used to blanket both the coordinate and tetrad invariance together. A fantastic article which covers this in detail is Stewart and Walker (1974) (unfortunately pay-walled), where they discuss the importance of the distinction.