Nothing.
No.
This app only asks for permission to write inhaler usage data to HealthKit. It cannot read any data or write any other data.
This app does not do anything with your data besides store it to Apple's HealthKit framework. You can learn more about HealthKit and the Health app on Apple's website.
The only time your data will leave your device is when HealthKit gets encrypted and then backed up to iCloud, or when you make an encrypted backup to iTunes. You can read more about how your HealthKit data gets backed up on Apple's website.