Recall the Work Energy theorem,

$W_{cons.}+W_{non-cons}...=\Delta KE$

This relation tells us that whenever the net work done on a body is $+ve$, the body gains kinetic energy.

Notice that I used the term 'net' here. It doesn't matter what the work done by a particular force is, its the net which matters. A force may do less $-ve$ work and another force may do more $+ve$ work, resulting in net $+ve$ work. The system gains kinetic energy due to this work. Its a pretty neat way to develop intuition about the signs of work.

In your example, work done by you is $+ve$ and $-ve$ by friction. Since, the box is moving, the net work comes out to be $+ve$, therefore the body gains kinetic energy.
	
The kinetic energy of the system has nothing to do with internal energy. ***Internal energy consists kinetic energy of the molecules inside the system with respect to its centre of mass.***

Since molecular motion is primarily a function of temperature, the internal energy is sometimes called 'thermal energy'.

*Total energy of the system* $(E)$ = U *(Thermal energy)* + $E_{Kinetic\space of\space system}$ + $E_{Potential\space of\space system}$  ....
	
As said before, internal energy includes the kinetic energy of the molecules with respect to the centre of mass. Slowing down the centre of mass won't have any effect on the internal energy. It remains the same.