Let's be clear on some things. Heat is not a type of energy that a system possesses. It is a process of energy transfer. It would not be right to define heat as kinetic energy of random motion of particles, because this implies that heat is a state function.
If a system was at a lower temperature than its surrounding, it's implied that there is an adiabatic wall between the system and the surrounding. If an exothermic reaction happened within the system, no heat would be transferred from the system to the surrounding; the temperature of the system would just increase. In most cases, though, the system and the surrounding would be at equal temperature (diathermic), so the exothermic reaction would result in heat flowing out of the system (which is now at a higher temperature due to the reaction) to the surrounding, raising the temperature of the surrounding. Think about bomb calorimetry as an example.