Another reason (perhaps the 'real' one?) for the capacitors, is to possibly help with rectifier reliability, a capacitor across them also helps to reduce any in-borne high voltage spikes that 'might' damage the diodes - however, rectifiers die with or without the capacitors fitted, and I've no idea if it helps or not.
I suspect the real reason is that it's 'been done for a very long time', and people just copy it - perhaps it's introduced as needed in University courses?, simply because of that reason. I've never designed anything with the capacitors across the rectifiers, and can't see myself ever doing so - it's just something else to go wrong (and they do go S/C occasionally), and appears to give no advantage.
The speaker thing is ludicrous - and you certainly won't hear any difference, and there won't be any difference, however some speaker manufacturers have occasionally fitted a wire between one side of the voice coil and the chassis of the speaker, presumably simply to ground the chassis. Possibly this is what has created such a myth?.