You want Sketcher Symmetry but it's not quite what other CAD packages have. Select what you want to mirror, which is probably the entire sketch using the drag selection box, select the Sketcher Symmetry
tool, then press J
to add constraints.
It's not perfect, I suspect because it can't read your mind as well as the big CAD packages about where you want coincident constraints for joining lines, but it's mostly OK.
EDIT: I can't tell if you are describing the Sketcher Symmetry tool in your 4th paragraph or not. You will sometimes get over constraints where the end points intersect but you can delete them with one click and make those points coincident instead. I suppose this could be time consuming if you have a really really complicated sketch with multiple intersecting points, but that probably wouldn't pad anyway.
I feel like this is not how it works for me. It does copy the constraints and automatically create symmetry between any points across the selected center line, but if you're making a mirror why is that not what you want?
Also I'm unsure about the behavior in older versions of freecad, the constraint behavior of this feature is new in 1.0