Metrics don't come over via Garmin Autosync, unfortunately, but they can be synced over https://www.fitnesssyncer.com/. Fitness Syncer will sync metrics at midnight, so it will essentially be a day behind (if you weigh yourself this morning, for example, it won't be in TrainingPeaks till after midnight). That said, the premium option allows for immediate syncing if that's something you're interested in.
Currently TrainingPeaks allows us to upload:
Weight
Water Percent
Percent Fat
Muscle Mass
BMI
From a quick look, the only one that Garmin provides us that does not get pushed over is Bone Mass.
This is interesting, I've been looking into this again recently and sent a query over to the FitnessSyncer people regarding what data from the Index makes it over to TrainingPeaks. They replied:
This sounds great if it's the case; could you confirm if these data all make it across to TrainingPeaks from your Garmin Index uploads?
Update: https://peaksware.uservoice.com/forums/106657-trainingpeaks-customer-feedback/suggestions/18549610-direct-garmin-connect-weight-steps-sleep-rest-hr-s