Gravitational interactions by energy or by mass? The mass-energy equivalence, first established by Einstein is an important and highly discussed phenomenon in physics. Without claiming much knowledge about high-end discussions on this topic, I would like to ask a question on this.   

Does the equivalence allow "energy" to exhibit gravitational
  interactions with other masses and energy?

Furthermore, how can an intangible thing like energy be localized in space which is necessary, atleast according to the classical view, for gravity? This could help solve a related question that 

does a matter-annihilation event between two particles also ceases the
  gravitational effect the original particles might be having on
  adjacent (and far-off) particles?  

I recently attended a lecture by Ashoke Sen where he probably hinted towards gravitational interaction by energy, but I might be mistaken as it was only a passing reference.
 A: On your first question: absolutely, energy gravitates (or induces curvature in spacetime) the same way that mass gravitates. If you read general relativity, you will learn that it is in fact the Stress-Energy Tensor that is the source of gravitational interaction (or equivalently spacetime curvature). Energy can be localized very easily; a parallel-plate capacitor has electrical energy localized between its parallel plates (and that electrical energy gravitates!). In fact celestial bodies have most of their masses owing to the energy of the bonds that keep their fundamental constituents (quarks) together in their atoms' nuclei. You can imagine any of these bonds as a spring connecting two fundamental constituents; every spring has potential energy localized on it and this potential energy looks like mass to macroscopic observers that do not see the tiny springs inside nuclei. In any case, thinking of the bond energy as mass or energy does not really matter; what matters is that the bonds (springs) contribute to the Stress-Energy Tensor, and the stress tensor gravitates.
The answer to your last question is negative. The energy is conserved in an annihilation process. Take the annihilation of a pair of electron-positron to form a photon as an example. The energy of the annihilating electron and anti-electron (positron) equals the energy of the resulting photon. Since energy gravitates, the out-going photon gravitates as much as the in-going electron and positron were gravitating.
