I'm wondering how to perform the operation on the selected vertex (left) that will result in the curve on the right:
Yes, those are loose edges. I tried the Bevel Tool and Ctrl+B, but they'll only operate in fully formed faces/edges.
I saw a teammate do this very operation with Blender 2.79 (I'm on 2.80) but he couldn't even name the operation, just a complex shortcut.
By the way: I'll want to transform this edge mesh in tubes, later!