Based on what you say, and what I read on the FreeCAD wiki about Sketcher ToggleConstruction, I understand that I want points/lines that are red/white, NOT blue. I believe this is what I had before. Nevertheless, see attached proof that my lines are all real, and my shapes are fully constrained:
I can't seem to get my vertices larger than 9pt; you can see that the 'enlarge' arrow is greyed-out. Even after lowering line thickness to 3pt, those vertices still are practically invisible. Here's what my settings window looks like:
When I try to use the additive pipe tool, some lines and vertices are selectable, but not the ones I need. It seems as though I can always select points that are on convex edges or flat surfaces, but never ones that are in concave edges. So, as shown, I can select this line on the flat surface, but not the overlapping line which is part of the desired sketch:
Find attached the version of this file that these screenshots were taken from.