Does האם actually mean the mom? I have never seen this before.
This is a rather high-level Hebrew word. However it is in use. For example, "Yom Ha'Em" is mother's day in Israel, not "Yom Ha'Ima".
In non-biblical hebrew, is there a difference between Mem and Final Mem?
No, they are pronounced the same.
The pronounciation is the same, but it's not written the same.
My translation is exactly as shown above ie.
האמא הואבת חלב ולחם
It. wasn't accepted.
How and why not?
You mixed up ה and א in your second word
Not this sentance, but before, I used the ampersand instead of and, but you said 'in correct