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.
gμν=ˆeμ⋅ˆeν
2. Like any other tensor. I.e. gμν is the covariant form, gμν is the contravariant form, and gνμ=gμν is the mixed form.
3. As I said earlier, it gives the dot product between the unit vectors. I.e.
gμν=ˆeμ⋅ˆeν=∥ˆeμ∥cosθ∥ˆeν∥
Note: in flat spacetime, this term would simplify a bit, obviously, to just cosθ.
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 ημν and a curved coordinate system gμν.
5. Just' like how x is different from 5x2+cossinlnx+icosx+2sin(x2). Riemann Curvature, Ricci Curvature, etc. can be written completely in terms of the metric tensor. E.g.
Γikℓ=12gim(∂gmk∂xℓ+∂gmℓ∂xk−∂gkℓ∂xm)=12gim(∂ℓgmk+∂kgmℓ−∂mgkℓ)=12gim(gmk,ℓ+gmℓ,k−gkℓ,m),
(christoffel symbols)
Rρσμν=∂μΓρνσ−∂νΓρμσ+ΓρμλΓλνσ−ΓρνλΓλμσ
(riemann curvature tensor)
Rμν=gρσRμνρσ
(ricci scalar)
R=gμνRμν
For a more detailed discussion, you may want to get a good GR text - book, like Ludvigsen General Relativity: A geometric approach.