Could spacetime itself be a field?
No, according to the current understanding acquired since the beginning of the 20th physics, spacetime is a mathematical model/environment where all physics takes place, in particular where particles and field 'live'.
Yes, in General Relativity, spacetime is a field described by a metric tensor.
I think that in GR I would associate spacetime with the whole Riemannian manifold, not just the metric.
No, it is a quality of the gravitational field but is not a field in it's own sense.
Spacetime is not a field. A spacetime is a pair (M,g) where M is a smooth manifold and g (the metric) is a tensor field on M. It's considered OK to refer to M as "spacetime", even though it would be more accurate to call it something like "spacetime's underlying manifold". If we use this terminology, we can say that the metric is a field on spacetime.
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