I did indeed get the sax from an online retailer - music123, which is connected to Musician's Friend. It was a risk, but I knew that.
The two octave keys do seem to open and close as Mr Howard described. That isn't to say that there aren't any leaks, but everything looks all right. I did dry the pads with some high-powed cleaning papers after playing this evening and will see if that makes any difference.
I didn't try the stock mouthpiece this evening, but I did play around with the ligature a bit. It might have been my imagination, but the multiphonic effect seemed to be less frequent when I loosened the ligature (a Rovner of some sort) and pushed it bugger back on the mouthpiece - so more of the reed was able to vibrate. Is that perhaps related to the odd noise...?
I also noticed today, before playing, that the inside of my lower lip was a bit sore, further down from the mouth than I thought it ought to be. So perhaps I in some way changed my embouchure and that accounted for the noise?
I now see that there are so many variables, and I suppose I might need to find a teacher for at least a while to ensure that I don't develop too many bad habits. The money is the problem there...