There is a statement about the 2nd Chern class $c_2$ concerning complex vector bundles: $$ 1/ (8 \pi^2) \int tr[F∧F] =c_2 \in \mathbb{Z} $$
(1) Does this statement depend on the choice of gauge group for the curvature 2-form (or field strength?), say in the case of SO(N), SU(N), or Sp(N)? How does this formula get modified for different gauge groups?
(2) Since the trace tr suggests that the we are writing the Lie algebra generators in terms of certain Representations. How do the Representations affect the expression of the above formula? How do we know which normalization to choose for certain Representations? [for example, choosing between Representations of fundamentals, adjoint or vector etc.]
(3) How does the formula above change regarding to the spin structures of manifold (spin or not)? [e.g. the normalization $1/ (8 \pi^2)$ changes. How and why?]
1) The second Chern class is an invariant associated to a complex vector bundle $E\rightarrow M$ over a manifold (spacetime) $M$ (or, equivalently, to a principal $U(n)$-bundle over $M$). It is for this reason that it depends on your choice of structure group (what you call gauge group) - its not even defined for $Sp(n)$ or $SO(n)$-bundles.
Now there are homomorphisms $Sp(n)\rightarrow U(2n)$ and $SO(n)\rightarrow U(n)$ which one may use to associate a complex vector (or principal) bundle to a given real or quaternionic bundle. After this one may ask for the Chern classes of the complexified bundle, and this may or may not give meaningful information about the original problem.
Observe, however, that $Sp(1)\cong SU(2)$, and this isomorphism sends the first quaternionic Pontrjagin class $q_1\in H^4BSp(1)$ to the second Chern class $c_2\in H^2BSU(2)$. Hence in the special case here it is meaningful to disuss the second Chern class of a rank 1 quatnerionic bundle. Moreover, if $M$ is of dimension $\leq 4$ then any $Sp(n)$-bundle will admit a unique reduction of structure to an $Sp(1)$-bundle, so again it is meaningful to ask for the second Chern class of a higher rank quaternionic bundle, and again what you are really getting is the second Chern class of the corresponding $SU(2)$-bundle. In this case again, what you get doesn't depend on your choice of $Sp(n)$- or $SU(n)$-structure group, but you should understand the full chain of isomorphisms you are invoking when writing this simple statement.
Likewise, if you ask for the second Chern class of a real bundle with complex structure, then what you are getting is the second Chern class of the associated complex bundle (and this is the only meaningful way you can ask for its Chern classes). Note however that the complex structure need not be unique, so the Chern classes you get depend on your choice of complex stucture.
2) The choice of representations is exactly the information specified by a complex or symplectic structure on a given bundle. These structures need not be unique, and correspond to different choices of representations. Different choices will generally yield different values for the Chern classes.
3) Spinor bundles have their own set of associated characteristic classes, namely the Steifel-Whitney and Pontrjagin classes, that come from their relationship with special orthogonal bundles. If you want to relate the Chern classes to spin structures then, at least in low dimensions, you will need to use one of the maps $Spin(3)\cong SU(2)$, $Sp(2)\cong Spin(4)\rightarrow Spin(5)\cong U(4)$. It's not so much that your normalisation will change, as you will need to recalculate your connection form on the restructured bundle, and take into account the representation you have chosen when altering the structure.