why hanashi wo suru toki and not simply hanasu toki
Hanashi is talk, story, conversation. The added ‘suru’ makes it talking, doing talk, having a talk. ‘When talking’ makes more grammatical sense than ‘when talk’. On the other hand, Hanashi chu would probably work and so would Hanasu toki, as you mentioned, (hanashi with a ‘U’ instead of ‘I’ to make it a verb). 話をする時 was just the author’s style in this example. Hanasu toki is acceptable.