if the pipes are cut from a Spanning Tree connecting all the hexagons (like the one you witness once you have solved), then yes, for those give pipe pieces, there should be only one solution.
Is this true? I can almost imagine one puzzle that can create two different spanning trees. Maybe I ought to draw it up to verify.
Spanning trees may be required but not sufficient for a unique solution.
You are right. It is possible that branches (pipes) cut from one Spanning Tree might be re-arrangeable into a different Spanning Tree.
All puzzles on the site have a unique solution At least my solver thinks so)).
Configurations with multiple solutions are possible, but I weeded those out when generating.