For your second kind of example, consider the foliation of the unit $3$-sphere that is the integral curves of the vector field $$ X = p\left(x^1\frac{\partial\ }{\partial x^0 } -x^0\frac{\partial\ }{\partial x^1 }\right) + q\left(x^2\frac{\partial\ }{\partial x^3 } -x^3\frac{\partial\ }{\partial x^2 }\right), $$ where $p$ and $q$ are relatively prime integers. This has a transverse metric, but it is not the fibers of any submersion from the $3$-sphere to a $2$-manifold.

As for things that don't admit transverse metrics at all, you want a foliation such that the holonomy of the leaves (actually, it's enough to have one such leaf) is not compact. A good example of this is the foliation of the unit circle bundle of a compact surface of negative curvature by the tangential lifts of geodesics of the metric.