Thermodynamics is the field of physics that deals with the relationship between heat and other properties (such as pressure, density, temperature, etc.) in a substance. Such processes usually result in work being done by the system and are guided by the laws of thermodynamics.
Equilibrium in thermodynamics refers to the situation when macroscopic variables describing the thermodynamic state of a system is not depended on time. Equilibrium of a system in mechanics means the net external force and torque on the system are zero. The temperature of a body is related to its average internal energy, not to the kinetic energy of motion of its centre of mass. A bullet fired from a gun is not at a higher temperature because of its high speed.
Heat capacity, in general, depends on the process of the system that goes through when the heat is supplied.
In a state of thermodynamic equilibrium, the microscopic constituents of a system are not in equilibrium.
In isothermal quasi-static processes, heat is absorbed or given out by the system even though at every stage the gas has the same temperature as that of the surrounding reservoir. This is possible because of the infinitesimal difference in temperature between the system and the reservoir.