Does the Ballet Crypto app store my Ballet wallet private key on my mobile device or transmit it over the internet?

No, the Ballet Crypto app never stores your wallet’s private key on the device. The BIP38 two-factor private key (composed of a private key entropy and a passphrase entropy) is stored on the physical Ballet wallet itself, and nowhere else. To send coins out from your Ballet wallet, you must enter the wallet passphrase entropy and scan the private key entropy with the app. When you do that, the private key is stored in the mobile device’s RAM for only a moment to cryptographically sign the transaction, then it is immediately deleted. It never leaves the secure environment of the app and it is never exposed to the internet.