Re:
You are correct. A nine speed set up uses the same freehub width, but the narrower sprocket spacing on the cassette requires a narrower nine speed chain and nine speed shifters to match up.
With chainrings it generally makes no difference whether they are 7, 8 or 9 speed chainrings. 10 and 11 speed rings are a little narrower but would still work with a nine speed chain.