What happens during mass-energy conversion? A mass is another form of energy. When a mass ceases to exist as 'matter', it exists as energy - in the forms of energy we generally know (light, heat). But is this so simple? When a mass exists in its usual form (a particle for instance), it creates a space time curvature. When the mass ceases to exist anymore in its usual, can it be not that that the curved space time returns to its initial state with the release of energy, - the energy that we actually obtain?
Taking an analogy to make the question clear --- When an object is placed on a stretched string, the string gets deformed storing some energy in it. When the particle is taken away, the string gets un-stretched with the releases of the energy. Can such an explanation not run for mass energy conversion?
I am a beginner enthusiastic in cosmology, please clarify to me.
 A: Whenever you have mass you have energy too, lots of energy for a tiny bit of mass.
And it is energy not mass, that is related to spacetime curvature. 
Your idea that mass curves spacetime and energy does not, is a lie, completely 100% baseless and simply untrue.
It's just that the energy associated with mass is the largest energy you are used to seeing every day, so when you ignore all other energy your results don't change very much.
It's like ignoring the spare change in your couch when talking about the nation's wealth. It's so small you aren't very wrong when you ignore it. But if you think that putting money into couches decreases the nation's wealth then you entirely misunderstand that it's all money (and all energy) that matters.
To answer your title question, inside the sun every single second mass energy of hydrogen is turned into other kinds of energy and it takes millions of years for that other energy to get out and it doesn't act any different until the energy actually escapes and leaves. When there is a certain amount of energy in the sun, we have a certain amount of curvature. Only when the energy leaves does something change. And the conversion of energy from one type to another has no gravitational effect.
As for cosmology, it's just like in the sun. Different forms of energy can move differently. And that's the only things that makes the situation change when mass is created or destroyed. The energy can move around differently when it converts to different forms.
A: According to Einstein's theory of general relativity, both matter and energy curve spacetime. The theory already makes an allowance for matter-energy equivalence. The Einstein field equations are: $$G_{\mu\nu} + \Lambda g_{\mu\nu} = k T_{\mu\nu}$$ The left hand side has the Einstein tensor G which encapsulates the curvature of spacetime, and the right side had T, the stress-energy or energy-momentum tensor, which includes mass and energy.
You also said:

Taking an analogy to make the question clear --- When an object is placed on a stretched string, the string gets deformed storing some energy in it. When the particle is taken away, the string gets un-stretched with the releases of the energy. Can such an explanation not run for mass energy conversion?

Let me emphasise that curved spacetime does not store energy inherently, and the curvature doesn't change when you (somehow) convert mass to energy. So your analogy is not accurate or correct. 
