I'm just starting my first max stat project and thought I'd better gather some info on nurturing for stats:
How much does nurturing boost a sylesti's stats by, and does it work on both eggs and hatchlings?

IIRC, its 5 nurtures for a single stat point increase. (Someone please correct me if I'm wrong on this number.)
Generally, the lowest stat is the one that will be given the single point increase.

Only nurturing hatchlings increases the stats, nurturing egglings speeds up hatching time.

It can be difficult to get a high level of nurtures outside of festival time so most stat projects (and certainly if you want max health) rely on philters (e.g. Prismatic Philter) as opposed to nurturing.