Can the Arduino interface with a fingerprint sensor on my mobile?
You'd need an app on your mobile to read the sensor and communicate the result to your Arduino in a secure way that cannot be spoofed by an intruder. That secure communication over an open channel (WiFi, I assume?) is not trivial to do right, so that would be a weak point in your security.

Blynk, an app+server+Arduino library combination provides an authentication token to pair with your Arduino in a relatively secure way. You quickly make (drag-and-drop) a control panel for your Arduino system with it. It doesn't offer fingerprint sensing directly, but if you keep your Blynk authentication token a secret, keep your Blynk app+token on your mobile, and the mobile is fingerprint-locked, you'd have essentially the same thing, and the pieces already exist. You'd only need to code the smart-home parts for the Arduino.