What is the formal definition of force field? Which is more fundamental force or field? Do field exist in nature (as force do i think as per section 12-1 of Feynman lecture volume 1, and page 8,9 of Keith Symon classical mechanics) or is it a mathematical definition? Why do field (take gravitational field for example), need time to propagate, what is it formally?
There is a name clash here. Vector fields are mathematical objects, functions mapping vectors to points in (3 or 4 dimensional) space. They are purely mathematical objects, existing by definition, independent from physical reality.
Meanwhile, Physical fields are believed to exist in nature. In the framework of classical mechanics, the introduction of newtonian gravitational field is avoidable, basically, you can freely decide if you want to work with instantenous forces acting between pairs of masses, or fields crested by mass distributions.
However, in modern physics, fields are seen as physically existing "things", a form of matter, capable of storing energy and propagating information. In some theories (for example Quantum Field Theory) all forms of matter (particles) are described as excitations of fields.
The confusion seems to arise from the fact that Vector (or scalar or tensor) fields are used as a model of physical fields. Even physicists often conflate them completely. For example, in full philosophical rigor, you cannot take the divergence of the electric field. You take the divergence of the vector field you use to model the electric field. But since all our observations suggest that the electric field can readily and fully modeled and understood using a vector field, nobody bothers to emphasize the distinction.
Regarding the propagation, it is not the mathematical consequence of their modelling as vector fields. You can, and indeed, in Newtonian mechanics or electrostatics often you do, use fields that react instantly, without needing time, to distant changes to configurations of its sources. So the mathematical formalism of finite speed propagation lies in the equations governing the time evolution of the fields. (Eg. Maxwell-equations for the EM field)
By the way, gravitational waves do not exist in Newtonian gravity. They are the consequence of the newer (102 years old) theory of general relativity. This theory uses an even more involved mathematical model, working not only with fields existing in space and time, but the curvature of spacetime itself.