Does it store my Wi-Fi password?

9:41192.168.1.107

Yes. The ticker keeps the network name and password in its own storage, which is how it rejoins by itself after a power cut. The password goes nowhere else: the price requests carry no such field, and the settings page never shows it. During setup the ticker paints the password on its own screen as asterisks unless you left the page’s eye button open. Device → Wi-Fi… → Erase Wi-Fi, or a factory reset, removes it. See Privacy.