1. It's a tensor that describes the geometry of spacetime (or any manifold, for that matter.). It's components give the dot products of the unit vectors in spacetime (or "vierbeins", or generally, "vielbeins".). I.e.

$$g_{\mu\nu}=\hat e_\mu\cdot \hat e_\nu$$

2. Like any other tensor. I.e. $g_{\mu\nu}$ is the covariant form, $g^{\mu\nu}$ is the contravariant form, and $g_\mu^\nu=g_\nu^\mu$ is the mixed form.

3. As I said earlier, it gives the dot product between the unit vectors. I.e.

$$g_{\mu\nu}=\hat e_\mu\cdot\hat e_\nu=\|\hat e_\mu\|\cos\theta\|\hat e_\nu\|$$

**Note: in flat spacetime, this term would simplify a bit, obviously, to just $\cos\theta$.**

4. As I said, the metric tensor can be expressed in terms of the vielbeins, so that answers the question. But speaking of coordiante systems, Christoffel symbols can map between say, a minkowski coordinate system with metric tensor $\eta_{\mu\nu}$ and a curved coordinate system $g_{\mu\nu}$.

5. Just' like how $x$ is different from $5x^{2+\cos\sin\ln x}+i\cos x+2\sin\left(x^2\right)$. Riemann Curvature, Ricci Curvature, etc. can be written completely in terms of the metric tensor. E.g.

$$\Gamma^i_{k\ell} = \frac{1}{2}g^{im} \left(\frac{\partial g_{mk}}{\partial x^\ell} + \frac{\partial g_{m\ell}}{\partial x^k} - \frac{\partial g_{k\ell}}{\partial x^m} \right) = {1 \over 2} g^{im} (\partial_\ell g_{mk} + \partial_k g_{m\ell} - \partial_m g_{k\ell}) = {1 \over 2} g^{im} (g_{mk,\ell} + g_{m\ell,k} - g_{k\ell,m}), $$ (christoffel symbols)

$$R^\rho{}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho{}_{\nu\sigma} - \partial_\nu\Gamma^\rho{}_{\mu\sigma} + \Gamma^\rho{}_{\mu\lambda}\Gamma^\lambda{}_{\nu\sigma} - \Gamma^\rho{}_{\nu\lambda}\Gamma^\lambda{}_{\mu\sigma}$$ (riemann curvature tensor)

$$R_{\mu\nu}=g^{\rho\sigma}R_{\mu\nu\rho\sigma} $$ (ricci scalar)

$$R=g^{\mu\nu}R_{\mu\nu}$$

For a more detailed discussion, you may want to get a good GR text - book, like Ludvigsen *General Relativity: A geometric approach*.