What I could understand is that particles are different kinds of local fluctuation states of force fields.
Particles are specific kinds of states (a certain asymptotic states defined according to the LSZ formalism) of the quantum fields--they do not necessarily correspond to the states of force fields. For example, the electromagnetic field is what can be called the "force field" responsible for mediating electromagnetic forces between electrically charged particles. And photons are the particle states corresponding to this field. But, electrons are also particle states of a quantum field, called the electron field. And the electron field is not responsible for mediating any force between any particles. Such fields are often called matter fields.
Particles make up matter. So matter is part of various force fields. Is that correct?
This is a bit of an oversimplification to say that particles make up matter. For example, the electron in an atom is not in a particle state. The key thing is that a quantum field has many physical states which are not necessarily particle states. Especially, if one is talking about strongly interacting fields such as the quark fields, there is no meaningful way in which one can talk of a "quark particle". However, it is correct that matter corresponds to states of various quantum fields--even if it is not made up of particles. Again, all quantum fields are not force fields. There are also the so-called matter fields such as the quantum field corresponding to the electron.
The metric of spacetime is the gravitational field. So is it correct
to say gravitational field is part of spacetime?
Yes, this is essentially correct. Gravitation is a manifestation of the intrinsic dynamical structure of the spacetime itself. However, notice that we do not have a quantum theory of gravity. And our notion of spacetime and gravity is likely to be massively revised at a conceptual level when we understand a full quantum theory of gravity.
What about other force fields? We have various force fields like
electromagnetic field, gluon field, higgs field,etc. Do these fields
lie on top of the spacetime (but separate from it) or are they too a
part of the spacetime?
No, such fields are additional structure on top of the intrinsic structure that spacetime has. Of course, they live on a spacetime manifold, i.e., a quantum field is simply a field of operators--one at each point in spacetime. But, the operator is an additional structure on top of spacetime.
However, let me mention an interesting point. In theories involving extra dimensions, such as the Kaluza-Klein theory, fields such as the electromagnetic field are not added on top of the basic geometrical structure of spacetime. Rather, they simply arise out of the geometric structure of spacetime--just like gravity does in general relativity.
Hope this helps! :)
