How is that after random numbers with their negatives alternating passing through 5, 4, 3 order moving average filter, become quasi sinusoidals? I have passed random numbers $${89, 58, 13, 70, 24}$$ with their negatives alternating, through 5 order moving average filter and output was positive and negative numbers alternating. After I have passed output through 4 order mov. average filter, output was positive and negative semi-periods! although no quasi-sinusoidal. But after passing this second output through 3 order moving average filter, output was quasi-sinusoidal! How then all that happened? 

Following are input signal and outputs.

Random numbers alternate with their negatives:

$${89, -89, 58, -58, 13, -13, 70, -70, 24, -24,
    89, -89, 58, -58, 13, -13, 70, -70, 24, -24,
    89, -89, 58, -58}$$

After passing through 5 order moving average:
$$2.6, -17.8, 14, -11.6, 4.8, -2.6, 17.8, -14, 11.6, -4.8, 2.6, -17.8, 14, -11.6, 4.8, -2.6, 17.8, -14, 11.6, -4.8$$

After passing through 4 order moving average: 
$$-3.2, -2.65, 1.15, 2.1, 1.5, 3.2, 2.65, -1.15, -2.1, -1.5, -3.2, -2.65, 1.15, 2.1, 1.5, 3.2, 2.65 $$

After passing through 3 order moving average: 

$$-1.56667, 0.2, 1.58333, 2.26667, 2.45, 1.56667, -0.2, -1.58333, -2.26667, -2.45, -1.56667, 0.2, 1.58333, 2.26667, 2.45$$
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.