"Will the torque remain constant while box is being pushed ?" Do you mean the torque due to gravity?

Measuring rest masses would give the same result at the start and at the end, measuring the relativistic masses - wont, because the relativistic mass doesn't take in account potential energy. So it goes: mass of the whole system - unchanged; relativistic mass - reduced; rest mass - unchanged; mass stored in potential energy - increased by the amount that relativistic mass was reduced.

You can also look at it like this: Joule is [kg * m^2 * s^-2], Newton is [kg * m * s^-2], so Joule is Newton*meter, so Energy is Force * Distance. This is kind of energy transfer is "Work" and you can see that if you haven't displaced the object (so Distance = 0) then the work you've done is also 0, so the energy you've transferred to the object is 0J.

Viewed from the wall/ball's perspective - they are not moving you, so their power output towards you is 0W, so they don't need to expend energy, because there is no power to keep up.

I'm going to be a bit over simplistic and say if there is no velocity there is no energy transfer. As long as you are not moving the ball/wall none of your energy goes into it. Look at the units: Energy - Joule, Power - Watt which is Joule/second, so Joule = Watt*second, so it takes energy to keep up power (NOT to keep up force). Watt on the other hand is Neuton * meter/second. If you are moving an object at 0m/s, your power output towards that object is 0W, so you can apply any force for any duration, as long as you're not moving it no energy goes into it.

As far as Newtonian mechanics are concerned Energy is preserved (Not Force). If a stationary ferromagnet is pulling a stationary piece of iron there is no total acceleration, so there is no power output, so no energy is expended, nothing gets "tiered". To address the "but nothing else does" imagine a car pressed head-on against a wall, if you step on the gas the car won't go anywhere, the tyres might spin in a burnout fashion and the car is obviously consuming energy, the car's components are accelerating/decelerating, but there is no overall forward-backward motion, just like your muscles.