My HAH powerhead for carpet is a "TurboCat." The website for that product says, "Will the Air Driven Power Brush clean as well as an Electric Powerhead? If the Turbocat Air Driven Power Brush is matched with the right Power Unit tests show that it�s performance is comparable to an Electric Power Head. The Turbocat has a higher life expectancy, higher brush speed than many electric power head�s, it is lighter, it has an automatic height adjustment feature, and it has the ability to clean closer to edges. In addition the Turbocat has a built in Performance Indicator. If airflow is restricted within the tubing or power unit, the Turbocat will slow down, indicating that the system is not operating at peak performance and should be checked." I have never used an electric powerhead to compare suction ability. My powerhead does a good enough job, but I imagine there is room for improvement....

Buy an extension cord with the polarized plug molded on it and cut the other end off. It is probably even cheaper than a plug and lamp cord purchased separately.

Ours is also woven through the chain. DD bought a house with a chandelier that had 3 support chains and the electric cord went straight up to the ceiling like yours. The previous owners didn't weave the wire through any of the support chains - instead they bought a new chain and wove the electric wire through that and then attached it to the ceiling - giving the chandelier a 4th chain.

Just make sure you resulting connections are parallel and not in series. I would pigtail all connection at the base of the lamp and then only run two conductors to each terminal so you don't have two wires under 1 terminal that could easily slip out. Use wirenuts for all splices and the wrap in electrical tape to keep everything secure and in place in case one of the cords gets yanked at.