What is the correct way of using "neither nor"?
Alex neither jumped nor sang.
Neither did she jump nor sing.
Neither jumped she nor sang.
is the usual format.
is framed more like a line from a poem, as though required for the metre and rhyme. While unusual, it's legitimate in poetry or verse:
With injured claw and wounded wing, neither did she jump nor sing.
I think expression 2 should be Neither did she jump or sing.
I don't think neither and nor are coordinated in your example 2.
The word neither attaches to the entire following phrase, and it must refer to something that came before, for example,
She didn't smile or laugh; neither did she jump or sing.