Macbook Mail plays incoming mail sound even after turning off all notifications
I have had a similar issue. I use SpamSieve and I was unaware (or had forgotten) that there is a preference within SpamSieve to make a sound after a message is identified as 'good'. That is why the Basso sound (the default from SpamSieve) was being played after the 'New Message' sound - that, in experimenting, I had chosen in Mail.

Are you using SpamSieve? Or perhaps another filtering service that rides on Mail?