Since $T$ is a closed current, it has a local primitive $u$, which is a function such that for any smooth $n - 1$-form $\varphi$ with support on the set where $u$ is defined,
$$\int_T \varphi = \int_M u ~d\varphi.$$
The mass of $T$ is exactly the total variation of $u$ and so your mass-minimizing assumptions amounts to asserting that $u$ minimizes its total variation, or in other words that $u$ is a function of least gradient, also known as a $1$-harmonic function.
Functions of least gradient have been extensively studied, especially for the Dirichlet problem on euclidean space. Górny and Mazón recently published a very nice monograph about functions of least gradient (which even tackles the case that you don't have a Riemannian metric, but only an elliptic integrand!) so I won't attempt to do the whole theory justice here and just summarize the key points.
Any function $u$ of least gradient has bounded variation, so we may apply the coarea formula
$$\int_M |Du| ~dV = \int_{-\infty}^\infty \mathrm{Per}(\{u > t\}) ~dt$$
(where $\mathrm{Per}$ denotes perimeter) to study its superlevel sets. It follows from the minimality that for each $t \in \mathbb R$, $\partial \{u > t\}$ is a mass-minimizing integral current. In particular it is smooth up to a set of codimension $8$. Górny has observed that therefore $u$ is locally in $L^\infty$, and he has shown many other regularity results about $u$ besides. I combined these facts to show that if $n \leq 7$ then the level sets of $u$ form a measured lamination of Lipschitz regularity. In other words, we can cover $M$ by an atlas of Lipschitz coordinate charts in which $u$ becomes $u(x, t) = f(t)$ for some function of bounded variation $f$ on an interval in $\mathbb R$.
From this last assertion it follows that if $n \leq 7$ then for any $n - 1$-form $\varphi$ which is supported in one of the coordinate charts,
$$\int_T \varphi = \int_K \int_{\Sigma_t} \varphi ~d\mu(t)$$
where $K$ is a compact subset of $\mathbb R$, $(\Sigma_t)_{t \in K}$ is a family of disjoint minimal hypersurfaces, and $\mu$ is a positive finite Borel measure with support $K$. The case that $\omega$ is integral is the case that $K$ is a finite set and $\mu$ is a finite sum of point masses. A weaker variant of this decomposition, which does not require $n \leq 7$, was observed by Auer and Bangert some years ago.
The example of the irrational foliation of the torus, that you already pointed out, shows that we can't really expect any regularity for $T$ itself (other than some regularity for its primitive $u$), but only the hypersurfaces that it decomposes into. On the other hand, the situation that the parameter space $K$ looks like a Cantor set (even a $0$-dimensional Cantor set!) cannot be avoided even in natural applications. This is what happens to the functions of least gradient which naturally arise in Teichmueller theory for example.
(For technical reasons, I might have accidentally assumed that $M$ is oriented in the above discussion. But I think that this isn't really a big deal, since for regularity questions you can just pass to the oriented double cover of $M$.)