-
The 9-speed chain has narrower side plates, so is narrower on the outside, but internally the widths of the links are almost identical.
Almost, but not quite. In practice, most cassette sprockets, yeah even unto 6-speed, have been stamped from 2mm stock so they will accept 9/10/11-speed chain, but 5/6/7/8-speed chain has a roller width of 2.4mm and 9/10/11-speed has a roller width of 2.2mm, so it's possible for a sprocket to exist which accepts one but not the other.
I've run several 9-speed chains on 7-speed cassettes and chainrings. They've all worked just fine. The 9-speed chain has narrower side plates, so is narrower on the outside, but internally the widths of the links are almost identical.
I don't know whether it would solve any chain dropping issues though.