Tips on how to install laminate over existing flooring

With it being such a short piece, you might try using some quarter-round and sanding the flat side of it down until the thickness matches that of the new floor. Since the floor is dark, perhaps just some dark caulk in the gap around the post would be sufficient. Are you planning to re-finish the stair post/balusters to match the new floor?

I would cut a 3/4" strip of flooring and use some construction adhesive to fasten it to the edge of the stair and install the stair nose up against it with some adhesive. That won't expand and contract like the flooring might. Then fit the perpendicular pieces snug up against the back edge of the nose or underneath it if it's designed to overlap a bit. You might need a drop or two of adhesive on a few of those members to hold them down at the edge of the nose. Good luck.