I want to move an edge of object A onto the face of object B. I want them to meet where the blue lock-axis meets the face of B. How can I do this?
You can enable Snapping by pressing the magnet at the top of the screen. If you click on the drop-down menu next to the magnet icon, you can change what Blender will snap to.
To toggle snapping via a keyboard shortcut, press Shift+Tab. Pressing Ctrl+Shift+Tab will bring up a pop-up menu for changing the snapping object.
If you don't want to toggle snapping, you can just hold Ctrl instead to enable snapping only while Ctrl is held down.
Correct me if i am wrong. It seems to me that the two edges you want to snap to eachoother are obsolete. The way it appears in the images you can remove those edgeloops, because the end result is a mesh that has the original shape of a cube, but it is broader in one end. Can you provide an image displaying the whole mesh?