In QFT, space is the same three-dimensional “Euclidean” space that we intuitively believe in, and time is the same time that we intuitively believe in.
The gravitational field is a force field like the other force fields, but with a higher spin, or helicity, of 2. Four-dimensional curvature is best left to the physicists who find it useful in their calculations.
General relativity is compatible with QFT, at least in Schwinger’s formulation. However, unlike the equations for the EM field, the gravity field equations cannot be renormalized and calculations cannot be made at the quantum level.