For $\bf{Q1}$, (local) congruence of sub-manifolds under a continuous group action can be determined by the method of moving (co)frames (inspired by Elie Cartan and rigorously formulated by Olver and Fels): see Section 10 in http://www.math.umn.edu/~olver/mf_/mcII.pdf.
For an application of this to image recognition and computer vision (since you mentioned it) see the section on image processing here: http://www.math.umn.edu/~olver/paper.html