Why are Hajime's attack names in French?
Almost certainly based on the Rule of Cool. Manga and anime frequently contain specialized jargon, names of attacks and locations in foreign languages simply because it sounds 'cool' and different to the Japanese ear: e.g. Spanish (Bleach), German (Neon Genesis Evangelion), English (most of them) or multiple (Full Metal Panic).

In Gatchaman, there may be an additional reason for using French for Ichinose's attacks specifically: to emphasize her femininity and artistic side. French is often associated with delicacy, grace, and the arts (e.g. ballet, couture, visual design, performance art, etc), and so may be a hint towards her "gentler nature" (chauvanism does still appear in anime). Her artistic side is also referenced by attacks related to her crafts hobbies, as you've mentioned.