How would normal matter behave under conditions found in the core of the sun? Considering how low the power density is at the sun's core, I seem not to be able to expect what would happen to matter in case it was thrown inside the sun's core.
For example, let's assume an Earth-like planet is placed at the center of the Sun's core, and with the power density of 276.5 $watts$/$m^3$ which is very low to even raise the temperature of the Earth by any noticeable degree. That was one thought.
The other thought was that, the temperature in the core is already 15 million degrees K, so any matter there should get close to this temperature quickly enough.
So now I'm very confused,  like will the planet stay there intact for thousands or even millions of years until it has accumulated enough energy to melt or vaporize ? 
Or there are other types of energy absorption the planet would experience and thus have a shorter time staying as one piece in the core ?
Just for trying to get a practical and numerical answer to this question, let it be : how long can the Earth as a whole survive in the core of the sun ?  
 A: As you say, the power produced per cubic metre of the Sun's core is surprisingly low. This is because proton-proton fusion is a very slow process, as has been discussed hereabouts before. The core is so hot because conduction of heat through the core is slow. The average speed with which a photon escapes the core is the astonishingly low value of about 30$\mu$m/s.
However the photon net speed is so slow because the dense plasma at the Sun's core scatters photon extremely efficiently. If you were to magic the Earth into the Sun's core then the Earth would start receiving energy at the rate predicted by the Stefan Boltzmann law. I make this around $10^{21}$ W/m$^2$ of the Earth's surface, so the Earth would start boiling away pretty quickly. The Earth would cool the plasma around it, and as that plasma cooled and recombined it would become transparent to the next layer of plasma out, and so on. To a first approximation the heat flux entering the Earth would remain at around $10^{21}$ W/m$^2$ until the material of the Earth became hot enough to form a plasma itself. At this point that plasma would start scattering photons and the heat flow would start slowing.
Actually calculating the rate at which the Earth would vaporise would be a difficult task. You could treat it as the heating of a sphere with a known thermal conductivity, but as mentioned above once the temperature gets hot enough to ionise the material from the Earth this would strongly affect the heat flow from the plasma around it.
A: Imagine you have a hot tub, and you heat it up to a nice toasty temperature.  Then the power goes out.  The metabolism of the hot tub environment is now zero.  It won't get any warmer.
But if you get in the tub, you'll still warm up.  The thermal mass of the water won't be cooled much by you entering.  You're taking advantage of the heat that was produced earlier.
The earth would be heated by the existing thermal mass of the core.  
A: If you were to "magically" place a planet in the sun's core, I'm fairly sure that is would not be there long. The ambient temperature of the sun's core is somewhere around 15.7 million K, as you said. You should think about why its so how there before you think about melting planets. The density of the core is something like 150 times denser than water. When you compress something that much, it gets hot. (Think of compressing a gas. What you compress a gas, it gets hot.) And when something like that is that compressed and that hot, a really huge fusion reaction occurs. So on one hand, placing an Earth inside the sun would increase the mass of the sun, negligibly increasing the pressure, increasing the heat a tiny bit. On the other hand, introducing something with so many heavy metals might do something to the fusion reaction of the sun (I don't know what, but it might.)
The appropriate question to ask yourself would involve the specific heat capacity of Earth, and the ambient core temperature of the sun. (Also, the melting point of Earth).
Though at 15.7 million K, I cannot think of anything that could stay in a solid form for long.
Just think of the boiling point of iron. (3134K) some simple division tells us that the sun is over 5000 times hot enough to boil iron. The iron would never even touch the sun before it boiled.
Another one, according to Wikipedia, granite melts at 1533K. More division tells us that the sun could melt granite more than 10,000 times over.
I hope I was able to answer your question, at least partially.
