This is irrelevant, since the researchers only claim that parents’ emotion when singing to their children affects the sound of the singing. The relative amount of emotion felt when singing to different audiences doesn’t affect that claim.

This isn’t relevant, because the psychologists were comparing recordings of the same parents singing with and without their children present. Any difference caused by being aware of the recording would affect both samples, so shouldn’t change the result.

The researchers don’t make any claims about emotion displayed by parents when singing in any other case than to their child, only that the emotion felt when parents are singing to their child noticeably affects the sound of their singing.

This strengthens by providing an mechanism to support the hypothesis. If we know how emotion can noticeably affect the sound of singing, it’s more plausible that results are truly due to parents’ emotion and not some other factor.

The researchers’ hypothesis doesn’t depend on the parents’ beliefs, only whether that is truly the case, so this is irrelevant. If anything, parents’ beliefs could lead them to sing differently, which could even weaken the hypothesis that emotion led to the differences.