If melting points and freezing points are the same, what state is pure equilibrium?
At the melting point there is a change of state. This means that solid is transformed in liquid or viceversa. For this process you need energy which is supplied or withdrawn in the form of heat. The whole process occurs at constant temperature, which is precisely the melting point. 

Adding or removing temperature has no sense. It is heat what  is added or removed. The temperature is just a result of the state. When you have a mixture of solid and liquid in equilibrium, this temperature is defined, is the same in the whole system and is the melting point. 

Since it is state change, both solid and liquid are present and the temperature will not change while both are there in any quantity and if the system is in equilibrium. The quantities of solid and liquid will not vary until there is some heat flow. However, on the microscopic level, the molecules are still moving and some will migrate from the solid to the liquid while others (the same amount) will go from the liquid to the solid. This results in that, even if the system is in equilibrium, the small crystals will dissolve while the bigger ones will grow bigger.