If you don't pay attention, I don't know that you'll learn anything here. Let's start with the temperature at the core. As the NASA article states, brown dwarf stars don't have a fusing core.
Brown dwarfs start their lives like stars, as collapsing balls of gas, but they lack the mass to burn nuclear fuel and radiate starlight.
I hope this time you get this point. I don't know what else I can do to help you correct your misunderstanding that repeating the above quote. There is no "ultra hot matter" (whatever that redundant term means...).
Sigh. Brown Dwarfs have a burning core. They burn Deuterium. It is a partial version of the Stellar Fusion process. I said nothing about Brown Dwarfs operating via fusion.
Next, let's deal with your confusion about convention. By definition, convention is the flow between two temperature differentials. You seem to assume that all conventions immediately lead to equalized temperatures throughout the star. There is no basis for that assumption. Clearly the surface radiates energy to space, including to the Spitzer and WISE. I encourage you to be on better guard against unwarranted assumptions.
Look at the illustration again. Matter is continuously recycling with the core. This is how it is displayed for Brown Dwarfs and Red Dwarfs. This is how astronomy says they work.
I tried the campfire example to assist you. Now let me have you imagine a lake that's frozen over. Why doesn't every lake with a frozen surface immediately become solid ice? Surely there's convention within the liquid water of the lake. Polar oceans on Earth have had frozen surfaces for eons, yet they are not, for the most part, not solid ice. So if you had reflected on your convention assumption for even a few minutes, you would have found your mistake on your own., Now I hope you can understand your error and accept that 0 degrees Celsius is, while unusual, not an improbable surface temperature of a brown dwarf. I hope that helps.
If the ice surface of a frozen-over lake were continuously brought to the bottom of the lake, it would no longer be ice. The entire system would reach an equilibrium.
You've added another, rather silly, claim that deuterium burning is not burning nuclear fuel. It is. And you've seen the NASA article quote. There's no nuclear fuel burnt in brown dwarfs, so there is no deuterium burning.
Where did I say anything about burning nuclear fuel? 
I haven't even written the word nuclear, or nuclear fuel, in this thread.
I seem unable to help you with more than one point at a time. Let's deal with nuclear first. Once you understand your mistake on this topic then maybe we can move on to your other mistakes.
The "burning" of deuterium must be a nuclear process. Deuterium is the nuclear fuel when deuterium is burned. So when you wrote "burning of deuterium" you wrote about nuclear fusion. It's that simple.
You may not understand that "burning of deuterium", a isotope of hydrogen, is not like the burning of charcoal. That's a chemical process, called oxidation. Molecular oxygen (typically) from the Earth's atmosphere combines with the longer carbon chains of the charcoal producing ash, heat, and carbon dioxide. Deuterium burning is the fusing of the nuclei of that isotope to form typically an isotope of helium.
[qoute author=http://en.wikipedia.org/wiki/Deuterium_burning]Deuterium burning is a nuclear fusion reaction that occurs in stars and some substellar objects, in which a deuterium nucleus and a proton combine to form a helium-3 nucleus. It occurs as the second stage of the proton–proton chain reaction, in which a deuterium nucleus formed from two protons fuses with a further proton, but can also proceed from primordial deuterium.[/quote]
Yes, brown dwarfs are hot enough to burn the second heaviest hydrogen isotope. No, they are not hot enough to burn the lighter, more abundant lighter isotope. No, they do not have a stellar core that produces light, so they can get their energy out to the surface by only convention, not radiation. No, brown dwarfs have no "second stage" deuterium burning, as only primordial deuterium is burnt in brown stars.
If you understand that the heat from a brown dwarf comes from only this nuclear fuel, in very limited supply, not very productive, not very efficient and producing nothing even close to the stellar furnaces of shining stars, I think we can move on to your confusion about the efficiency and speed of convention.