Both answers are "No."

There are well-known obstructions to the existence of an equivariant momentum mapping arising from the action by symplectomorphisms of a group $G$ on a symplectic manifold.  They can be phrased in many ways, but if $G$ is connected and its Lie algebra is semisimple, for example, the obstructions vanish.

A nice treatment can be found in the classic paper by Atiyah and Bott: "[The moment map and equivariant cohomology][1]" in Topology 1984.


  [1]: http://www.ams.org/mathscinet-getitem?mr=721448