Is virtual particle the same thing as force particle? Is virtual particle the same thing as force particle? Which of the above categories do photon, graviton and gluon fall into? Virtual particle, force particle, or both?
 A: 
Is virtual particle the same thing as force particle?

In a general sense , yes. But in terms of fundamental forces there is a distinction on which particles are fundamental force carriers
Force is a concept defined in classical dimensions as dp/dt, where p is the momentum .
A virtual particle is a quantum mechanical term, pertinent to the Feymman diagrams which describe the interactions of elementary and composite particles in quantum mechanical dimensions. Quantum mechanical dimensions are commensurate to h_bar. Virtual particles are the internal lines in the iconal representation of the diagrams, they carry the quantum numbers of their named particles but are off mass shell.

Feynman diagrams are graphical ways to represent exchange forces. Each point at which lines come together is called a vertex, and at each vertex one may examine the conservation laws which govern particle interactions. Each vertex must conserve charge, baryon number and lepton number

....


Particles are represented by lines with arrows to denote the direction of their travel, with antiparticles having their arrows reversed. Virtual particles are represented by wavy or broken lines and have no arrows.

All interactions between initial and final states carry a change in momentum, and thus a dp/dt. In this sense all virtual particles ( and these can be all elementary and composite quantum mechanical particles) transfer a force, a dp/dt.
You ask:

Which of the above categories do photon, graviton and gluon fall into? Virtual particle, force particle, or both?

The photon the Z,W and the gluon are called fundamental force carriers because their virtual exchanges dominate the first order diagrams in the electromagnetic, weak, and strong interactions respectively.
The photon when in an internal line as well as the  W,Z are virtual, force carriers,  but also can be on mass shell,  real particles. The gluon due to the nature of the strong interaction is always in internal lines and thus always virtual. The graviton is a hypothetical particle in effective quantizations of gravity in which it behaves for gravitational interactions/forces the same role as the photon for electromagnetic ones.
In summary, in the feynman diagrams all elementary (and composite) quantum mechanical particles can be virtual and thus force carriers in an interaction, but  the particles, called gauge bosons, photons, Z/W,gluon, graviton are associated with the respective four forces, electromagnetic, weak,strong, and gravitational, due to the strength of the first order diagrams .
