= Solution
Let $\mathcal J=\star J$ be the current $(d-1)$-form on an oriented $d$-dimensional spacetime. For a slab bounded by two spacelike hypersurfaces $\Sigma_1,\Sigma_2$ and a side boundary $B$, the <Stokes theorem> gives
$$
0=\int_Vd\mathcal J=\int_{\Sigma_2}\mathcal J-\int_{\Sigma_1}\mathcal J+\int_B\mathcal J.
$$
Thus $Q(\Sigma)=\int_\Sigma\mathcal J$ is conserved if the side flux vanishes, for example for a spatially compact current or appropriate falloff at infinity. With side flux it instead obeys the corresponding charge-balance equation. This is the differential-form version of current conservation.
For the <Abelian Chern--Simons theory>, take $A\mapsto A+d\chi$ with a smooth compactly supported gauge parameter. The change in the gauge-field term is a boundary term because $d\chi\wedge dA=d(\chi\,dA)$. For the source,
$$
\mathcal J\wedge d\chi=d(\chi\mathcal J)-\chi\,d\mathcal J.
$$
Discarding the boundary terms, the action variation is
$$
\delta_\chi S=c\int\chi\,d\mathcal J.
$$
<Gauge invariance> for every such $\chi$ and $c\ne0$ therefore gives \b[$d\star J=0$]. If $c=0$, the current decouples and <gauge invariance> places no condition on it. On a spacetime with boundary, boundary conditions or boundary degrees of freedom are also needed to handle the discarded terms; on a nontrivial compact gauge bundle, large-gauge invariance is an additional global issue.
For the field equation, use
$$
\delta(A\wedge dA)=2\delta A\wedge dA-d(A\wedge\delta A).
$$
The <one-form> $\delta A$ commutes with the <two-form> $\mathcal J$ under the <wedge product>, so
$$
\delta S=\int\delta A\wedge(2dA-c\mathcal J)
$$
up to the boundary term. Consequently
$$
\boxed{2dA=c\star J.}
$$
Applying $d$ and using $d^2=0$ again gives current conservation for $c\ne0$. The factor two comes from varying both occurrences of $A$.
Orient a spatial region $D$ so its boundary is $\gamma$, and define $Q_D=\int_D\mathcal J$. Applying <Stokes theorem> and the field equation yields
$$
\boxed{\Phi=\oint_\gamma A=\int_DdA=\frac c2Q_D.}
$$
For a <U(1) connection>, in a convention with unit minimal charge, the gauge-invariant quantity is its <holonomy> $e^{i\Phi}$, the Wilson-loop or Aharonov-Bohm phase. A large <gauge transformation> can change the chosen representative of $\Phi$ by $2\pi n$. Therefore \b[$\Phi$ is a phase angle modulo $2\pi$, rather than an absolute gauge-invariant real number]. A particle of charge $q$ has phase $e^{iq\Phi}$ in the corresponding normalization. The flux-charge relation thus attaches a gauge phase to enclosed charge.
The metric variation requires specifying the independent source. The pure Chern-Simons term $A\wedge dA$ has no metric dependence, hence contributes zero <stress-energy tensor>. If the <one-forms> named in the question have fixed covariant components $A_\mu,J_\mu$, the <Hodge star> in the source does depend on the metric:
$$
S_{\mathrm{source}}=-c\int d^3x\,\sqrt{-g}\,g^{\rho\sigma}J_\rho A_\sigma.
$$
Using $\delta\sqrt{-g}=-\sqrt{-g}g_{\mu\nu}\delta g^{\mu\nu}/2$ gives
$$
\delta S_{\mathrm{source}}
=-c\int d^3x\,\sqrt{-g}\left[J_{(\mu}A_{\nu)}-\frac12g_{\mu\nu}J_\rho A^\rho\right]\delta g^{\mu\nu}.
$$
Under that literal fixed-one-form convention,
$$
\boxed{T_{\mu\nu}=2cJ_{(\mu}A_{\nu)}-cg_{\mu\nu}J_\rho A^\rho.}
$$
There is another common convention in the topological source theory: hold the conserved <two-form> $\mathcal J=\star J$, equivalently the vector current density, fixed as the metric varies. Then both $A\wedge dA$ and $\mathcal J\wedge A$ are metric-independent, and \b[$T_{\mu\nu}=0$] for this action. These are different variations, not contradictory calculations. The <Chern-Simons source stress convention> explains why a source prescription is necessary; a dynamical matter source would contribute its own action and stress as well.
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