Identify whether each of the following is a former of mechanical energy: elastic potential energy, chemical potential energy, gravitational potential energy

I would say Elastic Potential Energy is a form of Mechanical Energy, because there is an associated motion with it.

I would also say Gravitational Potential Energy is a form of Mechanical Energy.

Chemical potential energy is not a form of Mechanical Energy, it is actually a form of Chemical Energy.  It is actually the energy contained in the chemical nature of a fluid.