Thank you Mark for the reference. It would be interesting, though, if for the case of Question 2 or the fact that $M$ is a fibre bundle over the circle with toric fibers would give a positive answer in the smooth catefgory, too.

It is the usual Lie derivative of a tensor

Thank you for your anwer. Fortunately I am interested in codimension 1 foliations on compact manifolds. In light of this, I found the following (maybe classical) result by Morgan: [Mor76]. It remains the question for me (being no expert in foliation theory) if there are maybe geometric or topological requirements on the manifold to have finite holonomy or the stronger holonomy restriction in Morgans paper cited above (Theorem 4 therein). I would appreciate any comment on this.