As the cohomology of $(S^1)^n$ is torsion free every stable bundle on $(S^1)^n$ is
determined by Chern classes (this also follows from the $K$-theory Künneth
formula) so just as for the spheres it is an unstable problem. As for the
unstable problem unless I have miscalculated, if $(S^1)^5\rightarrow S^5$ is a
degree $1$ map, then the pullback of the non-trivial $U(2)$ bundle on $S^5$ with
trivial Chern class is non-trivial. (The proof uses that the $5$'th step in the
Postnikov tower of $\mathrm{BU}(2)$ is a principal fibration $K(\mathbb
Z/2,5)\rightarrow U\rightarrow K(\mathbb Z,4)\times K(\mathbb Z,2)$.)
Some more details of the calculation: The first and second Chern class gives a
map
$$\mathrm{BU}(2)\rightarrow K((\mathbb Z,4)\times K(\mathbb Z,2)$$
which induces an isomorphism on homotopy groups in degrees up to $4$. As
$\pi_i(\mathrm{BU}(2))=\pi_{i-1}(\mathrm{SU}(2))$ for $i>2$ we get that
$\pi_5(\mathrm{BU}(2))=\pi_4(S^3)=\mathbb Z/2$. Hence, the next step $U$ in the
Postnikov tower of $\mathrm{BU}(2)$ is the pullback of the path space fibration
of a morphism $K(\mathbb Z,4)\times K(\mathbb Z,2)\rightarrow K(\mathbb Z/2,6)$.
In particular we have a principal fibration
$$K(\mathbb Z/2,5)\rightarrow
U\rightarrow K(\mathbb Z,4)\times K(\mathbb Z,2).$$
This means that for any space
$X$, the image of $[X,K(\mathbb Z/2,5)]$ in $[X,U]$ is in bijection with the
cokernel of $[X,K(\mathbb Z,3)\times K(\mathbb Z,1)]\rightarrow[X,K(\mathbb
Z/2,5)]$ obtained by applying $[X,-]$ to the looping of the structure map
$K(\mathbb Z,4)\times K(\mathbb Z,2)\rightarrow K(\mathbb Z/2,6)$. As
$H^4(K(\mathbb Z,3),\mathbb Z/2)=0$ the Künneth formula shows that any map
$K(\mathbb Z,3)\times K(\mathbb Z,1)\rightarrow K(\mathbb Z/2,5)$ factors
through the projection $K(\mathbb Z,3)\times K(\mathbb Z,1)\rightarrow K(\mathbb
Z,3)$ and $H^5(K(\mathbb Z,3),\mathbb Z/2)=\mathbb Z/2\mathrm{Sq}^2\rho\iota$ (where
$\iota$ is the canonical class, $\iota\in H^3(K(\mathbb Z,3),\mathbb Z)$ and
$\rho$ is induced by the reduction $\mathbb Z\rightarrow\mathbb Z/2$). Hence,
the map $[X,K(\mathbb Z,3)\times K(\mathbb Z,1)]\rightarrow[X,K(\mathbb
Z/2,5)]$ is either the zero map or given by the composite of the projection to
$H^3(X,\mathbb Z)$, the reduction to $\mathbb Z/2$ coefficients and
$\mathrm{Sq}^2$ (I actually think it is non-zero as otherwise the cohomology of
$H^\ast(\mathrm{BU}(2),\mathbb Z)$ would have $2$-torsion). If we apply it to
$X=(S^1)^5$ we get that $[X,K(\mathbb Z,3)\times K(\mathbb Z,1)]\rightarrow[X,K(\mathbb
Z/2,5)]$ is zero provided that
$$\mathrm{Sq}^2\colon H^3((S^1)^5,\mathbb
Z/2)\rightarrow H^5((S^1)^5,\mathbb
Z/2)$$
is zero. However, all Steenrod squares are zero on all of
$H^*((S^1)^n,\mathbb Z/2)$. Indeed, the Künneth and Cartan formulas reduce this to
$n=1$ where it is obvious.
A real-orientable ring spectrum $F$ admits a ring map from $MO$, and there is a straightforward ring map $MU\to MO$, so $F$ is also complex orientable. Moreover, $MO$ is a wedge of $H/2$'s, so $MO\wedge F$ is also a wedge of $H/2$'s, and $F$ is a retract of $MO\wedge F$ (by using the ring structure etc) so it is again a wedge of $H/2$'s. Thus, you don't expect to learn anything about (non)immersion from $F$ that you could not already learn from $H/2$ (although some kinds of bookkeeping may be simplified).
For the follow-up question:
The mapping spectrum $E=F(S^1_+,MU)$ is complex-orientable but not real-orientable and has
$$ \tilde{E}^1(\mathbb{R}P^\infty) =
\tilde{MU}^1(\mathbb{R}P^\infty) \oplus \tilde{MU}^0(\mathbb{R}P^\infty)
$$
Here $MU^{\ast}(\mathbb{R}P^\infty)=MU^*[[x]]/{[2]}(x)$ with $|x|=2$ and $MU^{-2}=MU_2\simeq\mathbb{Z}$ so the first summand is zero but the second is not.
Best Answer
Small correction: For a non-compact manifold $M$, the group $H_{2k-1}(M)$ might not be finitely generated. In this case Chern-Weil does not imply that the $k$th Chern class of a flat bundle on $M$ has finite order. Rather, it just implies that it belongs to the subgroup $Ext(H_{2k-1}(M),\mathbb Z)\subset H^{2k}(M)$.
Positive answer for first Chern class: Use the surjection $Hom(H_1(M),GL_1(\mathbb C))\to Ext(H_1(M),\mathbb Z)$ associated to the exponential exact sequence $0\to \mathbb Z\to \mathbb C\to GL_1(\mathbb C))\to 1$. An element of this $Hom$ group describes a flat complex line bundle on $M$ with prescribed Chern class in the $Ext$ part of $H^2(M)$.
Negative answer in general, for a pretty trivial reason: If $M$ is simply connected, then flat bundles on $M$ are necessarily trivial, but $M$ can still have torsion in $H^{2k}$ if $k>1$.
So a better question is, if $\Gamma$ is a group then can every element of (the $Ext$ part of ?) $H^{2k}(B\Gamma)$ be $c_k$ of a vector bundle arising from a homomorphism $\Gamma\to GL_r(\mathbb C)$ for some $r$? I don't know the answer if $k>1$.