I wish to ask whether I understand the following correctly. This universe seems to have six fundamental elementary bosons namely photon $(\gamma),\ W$-bosons$(W^+,W^-),$ gluon$(g),\ Z$-boson $(Z)$, Higgs boson$(H^0)$ and graviton$(G)$. Each fundamental force field of nature is mediated by the interaction of each of these particles, namely electromagnetism from $\gamma$, strong force from $g$, weak force from $ Z,W^+,W^-,$ gravitation from $G$. This brings forward another question, I know that particles get there mass from interaction of $H^0$, but does it also analogously result in some fundamental force ? If not, then why the exception?
While all the particles you mention are bosons they don't all play the same role since they have different spin. The photon, the $W$-bosons, the $Z$-boson and the gluons are all spin-1 particles. A force mediated by a spin-1 particle can be both attractive and repulsive depending on the charges of the particles exchanging the bosons. The graviton, on the other hand, has spin 2, and the force mediated by a pin-2 boson is always attractive. This explains why electric charges can either attract or repel each other depending on the signs of the charges, but gravity always makes masses attract to each other.
The Higgs boson is a scalar with spin 0. Like the spin-2 graviton a force mediated by a spin-0 scalar is always attractive. In the standard model the Higgs boson couples to the fermions through Yukawa couplings, which gives rise to interactions between them. The big mass of the Higgs boson means that the corresponding force is very short ranged, which means that it doesn't have any classical correspondence. However, the interactions between the Higgs boson and fermions is visible in a high energy experiment, where they are responsible for the processes that allowed LHC to detect the Higgs boson.