Could I just change shifters/cassette/derraileurs and leave the chainset as is?
Yes: chains only got narrower internally once, when nine speed was introduced (from 3/32" to 11/128"), so a ten speed chain (which you'll need for the cassette) will work fine with nine speed rings.
Yes: chains only got narrower internally once, when nine speed was introduced (from 3/32" to 11/128"), so a ten speed chain (which you'll need for the cassette) will work fine with nine speed rings.