Why only odd harmonics after non-linear amplification?
An ideal amplifier would have a transfer characteristic of $f(x)=Ax$: the input signal comes out amplified and otherwise undistorted. A real amplifier will deviate from this and go into saturation. We could model it by a polynomial $f(x) = \sum_{n} a_n x^n$. Now, what we would still expect is that the amplifier treats positive and negative values the same way, i.e., it is symmetric in the sense that $f(-x) = f(x)$. This leads to the requirement $a_n = 0$ for all even $n$. 

Now, one can show that raising a sinusoidal with frequency $f_0$ to the $n$-th power generates harmonics up to order $n$, however, preserving their oddity:


$n = 2$ generates frequencies 0 and $2f_0$
$n = 3$ generates frequencies $f_0$ and $3f_0$
$n = 4$ generates frequencies 0, $2f_0$ and $4f_0$
$n = 5$ generates frequencies $f_0$, $3f_0$ and $5f_0$


and so on. It's a consequence of the binomial theorem and quite easy to show (see, e.g., here).

In short: A transfer function that is odd symmetric will generate only odd harmonics. This is why you are seeing only those.