When is mechanical energy ever conserved for inelastic collisions?

Like in this https://www.khanacademy.org/science/physics/linear-momentum/elastic-and-inelastic-collisions/a/what-are-elastic-and-inelastic-collisions, in an inelasatic collision,the particles do not regain their shape and size completely after collision .Some fraction of the mechanical energy is retained by the colliding particles in the form of deformation potential energy.Thus,Kinetic energy of the particles no longer remains conserved anf gets converted into something else which is usually heat.However in absence of external forces,law of conservation of linear momentum still holds good.

Collisions are said to be perfectly elastic if the particles stick together and move with same velocity.Momentum is of course conserved here but maximum kinetic energy is lost in this case.

Like for example take a bullet getting stuck on a block hanging on a spring.When the bullet hits the block then the bullet and block would become hot. This is still a form of kinetic energy in reality, but it's now the random jiggling of molecules. This is not considered mechanical energy since it is not easily used for mechanical purposes.

I was uncertain of the definition of the term mechanical energy.

The mechanical energy is defined as the sum of both the kinetic and potential energy of a system.

The second part of the question asked whether or not mechanical energy is conserved if the system is earth and the two objects colliding perfectly inelastically afterwards.

As you correctly pointed out, after the inelastic collision the kinetic energy transformed into heat energy. If you regard your system as being the Earth and the two colliding objects (which are stuck), there is no energy crossing the system boundary. This means that the energy of the system is conserved.

However, the mechanical energy is not conserved due to the fact that the kinetic energy is not the same before and after the collision. This fact implies that mechanical energy is not the same before and after the collision (I am working with 1D collisions so that potential energy is zero before and after the collision). Note that KE transformed into heat energy, which is not included into the mechanical energy definition.

The loss of mechanical energy is not dependent on the setting.

Total energy is conserved in collisions. In elastic collisions the combined KE of the colliding objects remains unchanged by the collision. In an inelastic collision some of the kinetic energy of the colliding objects is converted into other forms of energy, such as sound and heat, so the KE is reduced.

The sound and heat energy is not classed as mechanical energy, because it is effectively dissipated and can no longer be used to do work.