Normal colorings of cubic graphs (part 1)
Some old results by Vladimir Müller, On colorings of graphs without short cycles, Discrete Math. 26 (1979) 165-176, imply that Question 3 has an affirmative answer. Although the auxiliary graph is not assured to be in some way as elegant or natural as in the case of line graphs or their squares. The easiest example that I can come up with has 13 times the order of the line graph.
This construction also gives an affirmative answer to Question 2.