How is that after random numbers with their negatives alternating passing through 5, 4, 3 order moving average filter, become quasi sinusoidals?
There isn't a mystery here. Your input signal isn't random; you chose 5 numbers at random, perhaps, but then you repeated the sequence of 5 numbers several times. This is a periodic signal. 

In continuous time, a periodic signal has a Fourier series representation; its spectrum contains discrete lines (i.e. sinusoidal tones). 

In a finite-length discrete-time signal, such as the one you proposed, you see a similar phenomenon. The spectrum of a signal that is periodic within its duration will have a line structure to it. 

Your moving averages are simple lowpass filters. As you apply more and more averages, you're filtering out the higher harmonics, leaving a combination of relatively fewer sinusoids. Thus, output of the cascade of moving average filters has a readily discernible structure as a sum of a relatively small number of sinusoids, as you described.