1) Don't understand that comment. I just used several standard fd schemes directly on Test Case 2.1) You like transforming coords. So change to coords that move with the chars.
2) Constrain dy and dt so that you solve numerically along the chars. That’s what I’ve used in the past.
2) That is possible; however, it is not easy for me and it is difficult to generalise, seems to be the general consensus.