Understanding harmonics of different Amplifier Circuits (A,AB,AB w/OP-Amp)
Examine the output waveform, then think about what harmonics would be required to reproduce it. For example, here's what adding 66% of 3rd harmonic and 25% of 5th harmonic looks like:-



And here's the LTSpice FFT for that waveform:-



Just two odd harmonics with the right amplitude and phase were enough to create a waveform very similar to typical class B 'crossover' distortion. With the addition of a few more higher harmonics you could precisely match the output of your amp. What this means is that (in the frequency domain) your amplifier creates those harmonics. 

As you assumed, any deviation from a pure sine wave produces harmonics, and the greater the distortion the higher the harmonic content. You can also get some idea of which harmonics will be produced by examining the shape of the distorted waveform. 

Even harmonics only occur when the wave is non-symmetrical ie. positive and negative halves are different. Since a class B amp has symmetrical output it only produces odd harmonics. 

Sharper deviations produce higher harmonics. Class B crossover distortion puts a 'flat spot' in the middle of the waveform. This can be audible even when it is very small, because the sharp transition creates higher harmonics that the human ear can easily distinguish. And since a fixed amount of signal voltage is required to get past that 'flat spot', smaller signals suffer from higher distortion.  

Class A amplifiers have a smooth transition in the center but compress the wave shape at higher amplitudes, usually more on one half than the other. This creates low order even and odd harmonics that are less audible even at quite high distortion levels. It is also hard to make out in the waveform until the amount of distortion is very high, because our eyes are not good at seeing subtle deviations from a sine wave.     

NOTE: to get better FFT plots in LTSpice you should set a longer simulation time, eg. 1 second instead of 10ms. This will make the peaks sharper and reduce artifacting. You can also change the vertical scale to get rid of the noise floor, and make the frequency scale linear so the harmonics occur at equal distances.