I think you have the right idea. You can decompose the differential into three pieces (dependent on the metric). The leafwise part automatically squares to zero, so you can get leafwise cohomology. I suggest looking at the papers by Alvarez-Lopez and coauthors: "Dimension of the Leafwise Reduced Cohomology", "Hodge decomposition along the leaves of a Riemannian foliation", "Long time behavior of leafwise heat flow for Riemannian foliations". You do not need special metrics to define the leafwise cohomology, but when you have a bundle-like metric, you get extra results like the Hodge theorem.

I emphasize that my answer above refers to the set of leafwise differential forms as a subspace of the space of all differential forms $\Omega(M)$. If, however, you don't need to think of leafwise forms as living inside $\Omega(M)$, you can define the space of differential forms as mentioned before by @BertramArnold; the tangent bundle to the foliation is well-defined, and thus the dual of the tangent bundle is well defined, and so you can use that to generate leafwise differential forms (that are not in $\Omega(M)$). The construction above gives an isomorphism with a subspace of $\Omega(M)$.

Yes. The torus is just $\mathbb{C}$ mod a lattice. I was trying to think of another counterexample. All you would need would be to mod out a semi-simple algebraic group by a lattice subgroup, and then the result would have non trivial fundamental group and thus nontrivial $H_1$. But I really don't have much experience with semi-simple group actions. I know that would work for compact group actions.