Can a closed system be a non-equilibrium system? Can a system without any interaction with other system be a non-equilibrium system; or is it that gain and loss is always needed for non-equilibrium?
 A: Yes, of course. The simplest way to see this is to consider a system with non-equilibrium initial conditions. For example, if you set up a gas with a non-uniform concentration (the most extreme example would be to push all the gas on one side of the container and leave the other side empty), the system will evolve in time until it thermalises. In the mean time you get non-equilibrium dynamics in a closed system. Of course, you can always argue the this system is not isolated because the initial conditions require some external intervention.
There are also system that simply do not thermalise, even when the initial conditions are far away in the past. There the 'non-equilirium' is truly a property of the dynamics, not the initial conditions. Right now I can think of two set-ups where this can happens:


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*Glassy dynamics describe the dynamics of disordered many-body systems. Note that this is inspired by, but is not the dynamics of glass. Basically, the potential energy landscape of the system is extremely complex and the dynamics takes an astronomically long time to minimise its energy because it gets stuck in local minima. I am no expert in this topic and can not give an example of isolated glassy dynamics. But I don't see why this does not exist.

*Integrable dynamics describe systems with an infinite number of conserved quantities. In these systems, thermalisation is hampered because the equilibrium state is not compatible with all the conservation laws. These systems often arise for $1d$ and are usually 'solvable' is some way. They may seem highly theoretical, but experiments are catching up! See this (and this link for the free version). The buzzword that you are looking for in this context is 'Generalised Gibbs Ensemble'. Google it, you will find a lot of litterature.
Note that thermal equilibrium is actually not a straight forward thing to define. You can look into this S.E. answer of mine where I go into this in more detail.
A: Yes. An equilibrium system is one that can be described by equilibrium statistical mechanics. This usually means that it is in a steady state of some kind. Systems that are started out in a non-stable state will undergo dynamical evolution. This is called non-equilibrium. 
Some isolated systems tend to approach an equilibrium state after some time. This process is called thermalisation and e.g. occurs in heavy-ion collisions. There is a lot of active research in this area.
