Can something have energy without having momentum?

Yes, potential enertgy in Newtonian mechanics and rest energy in Relativity.

Above is right. I'd just like to add that energy transmitted by massless waves would have energy and no momentum since there is no mass. Gamma radiation is very energetic, but momentum is velocity x mass.

yes its called potential energy, the formula is (mgh) ie mass x gravity x height.

energy is never lost, just tranferred. so a stone dropped from a 10th storey building starts with PE (mgh) then as soon as it falls its converted into kinetic or moving energy (1/2 x m x v squared). then it will reach Vmax as it hits the ground and all its kinetic or moving energay will then be changed into sound energy, and heat energy, and retrospective kinetic 'bounce'

It can have potential energy, but not kinetic energy.

Kinetic energy is just the rate of change of the momentum.

Yes, Potential energy is defined as a work of certain force (say, gravitational force, or Coulomb force, etc) during change of the relative positions (configurations) of the objects within a physical system. Often certain position is termed as reference position (or reference state) and the work is calculated versus this position. Because the work is reversible thus has the potential to change the state of other objects around it (for example, the configuration or motion) it was called potential energy.

Various forms of energy can be grouped as potential energy. Each of these forms is associated with a particular kind of force work of which is termed as potential energy. For example, work of elastic force is called elastic potential energy; work of gravitational force is called gravitational potential energy, work of the Coulomb force is called electric potential energy; work of strong nuclear force or weak nuclear force acting on the baryon charge is called nuclear potential energy. Chemical potential energy is the work of Coulomb force on during rearrangement of mutual positions of electrons and nuclei in atoms and molecules. Thermal energy usually has two components (kinetic energy of random motion of particles and potential energy of their mutual positions).

Yeah, sure. What you're talking about its stored energy like iun a spring, battery, or fat cells. Good enough yes?

of course! gravitational potential, chemical potential, elastic potential, heat, light

i'd go into mass being energy itself, but i don't know enough about that subject