These are all good questions, and we can add also "what is energy?" and "what is space?". I find it all deeply mysterious. Perhaps the questions are more interesting than the answers. How about this one: "what is is?"
Energy is actually deeply connected with time, as it’s the measurable quantity that’s conserved because of the time-symmetric nature of the laws of physics.
For anyone who's wondering, this is a consequence of Noether's theorem: https://en.wikipedia.org/wiki/Noether%27s_theorem. Similarly, symmetry under spatial translations implies momentum conservation, and symmetry under spatial rotations implies angular momentum conservation.
