So, basically meaning “in addition to”, “as well as”, the phrase seems to (sure, clunkily) mean to me: do not mention, do not add anything further.
No?
“And need I mention…” or “and I needn’t mention…”
Or, a family staple, “and while we’re on the subject.”
To me it means more like “and I have not yet managed to mention X”.
“And that’s not even getting into…”
No. The phrase introduces extra information that reinforces a point; while noting that the point would have stood even without mentioning this extra information. “To say nothing of” is the exact same formulation.
100% agree with @Riemann.
Soupy Sales and Pookie:
Pookie: I went to the museum, and it was stimulating, not to mention exhilarating!
Soupy: Exhilarating?
Pookie: I told you not to mention exhilarating!
From this cute clip, at 2:08 to 2:18 
The music, BTW, appears to be from this record by Clark Terry:
This is an example of an idiom. You cannot derive the meaning of an idiom from the dictionary definition of the words involved.
@Riemann explains the phrase perfectly.
Concurred, and thanks.
Only if you’re setting up a joke.
A-hyuk hyuk hyuk hyuk, yeah totally.