Heat is definitely not matter. However, the question of whether light is matter has been found throughout history as a problem - independent experiments have proven that light can be both a particle (called a photon), and an electromagnetic wave. This is a basic principle of quantum physics, called particle-wave duality.
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Five things that aren't matter are: sound, light, heat, gravity, and speed. Five things that are matter are air, water, foods, pencils, and paper.
Density (:
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Sun
No. At least, all available evidence seems to indicate that it is not possible to transfer matter, energy, or information faster than the speed of light.