The sun doesn't produce a lot of energy because it is hot. The sun produces energy because it is big. All functional experimental fusion reactors operate at much higher temperatures than the sun.
It's a straight up factually incorrect title. Heat is a measure of energy. The sun produces 3.9e26 joules of heat per second. So no, they did not surpass the heat of the sun.
Nah, that's a nitpick. In common English heat is just presence of high temperatures, and there's no specific word for thermal energy.
Also, that would be heat/time, so power basically. The sum of all heat currently in the sun would be larger, considering how slow the convection is among other things.
Heat in terms of temperature is a measure of molecular kinetic energy, so you can say two things are the same temperature regardless of the total energy of the system, which is a function of the change in temperature, mass and specific heat of a substance.
Even if you get as hot as the core, you're not doing the same amount of fission unless you have similar pressure at the same time, and no you don't. It's a bit complicated by the fact the sun is mostly protium, and has low volumetric energy output as a result, but that's the gist.