Is class A better at low volume?

Yes!!!

Not necessarily. Most Class AB SS amps are biased to a degree into Class A. So it depends on how the manufacturer biases the amp & your definition of "low volume."

Many class A amps are class A for the first 10% or 10 watts. After that they switch to A/B. Some amps are purely class A and won't switch. The easiest way to know is the amps power rating. If it is only a 30 watt amp (ie Pass or Threshold) it is most likely pure A. If it is higher rated, it probably will switch when pushed hard (aragon 200 watt?).

In addition to what Elevick correctly points out, some "Class A" amps drop into Class AB as the speaker impedance drops at certain volume levels.

In theory YES - high order harmonics can be created by nonlinearties in switching and bias. Switching crossover distortion tends to remain fixed and independent of signal level - therefore if the audio signal is very low then Class A will have lower "crossover" distortion than Class B or AB (as there is NO crossover). However in practice there are a great many distortion mechanisms in amplifiers and it it does not make sense to focus on just eliminating only one. (for example a a Class A may be less effective at higher output signals)

In practice a Class AB can be built quite well (bias reduce crossover distortion to a minimum) but some designs run in Class A mode until two thirds power to try to get the best of both worlds (attempt to be good at low levels AND at high levels).

Douglas Self on Power Amp Design