Question
when a meem/shaddah follows a tanween or a sakinnah letter, what exactly happens to the noon sound of tanween or noon sakin.

My husband says that you keep the noon sound of the tanwin.  I say that the tanween is completely hidden and  blends with the meem, where you do not even hear the noon, rather you make a Ghunna on the first letter of the meem and read the second meem with the vowel that come with it.

Answer
When a noon saakinah or tanween is followed by a meem, the noon saakinah merges into the meem and only the meem is heard, there is an accompanying ghunnah with this merging (idghaam). 
Please see:
http://www.abouttajweed.com/the-noon-saakinah-and-tanween/merging-of-the-noon-saakinah-and-tanween-p.html