I’m glad you’re asking mate, that’s how you learn! So capacitors typically do act as ‘batteries’ but in this case it is used as a coupling capacitor which more or less means it improves the response of the speaker.
As for the transistor, when the arduino makes pin 7 go from low(0V) to high (5V) it turns on the transistor which means that current will flow through the relay coil turning it on which completes the safety switch circuit. Now the transistor is used because the arduino can’t directly supply enough current to turn on the relay itself (from memory I think each arduino pin can only supply 20mA whereas some relays need more than this). The diode in this case is used as a flyback diode, the link explains it pretty well. Without it you could end up damaging the transistor.
All the grounds can just be connected together and to a single GND pin on the arduino, but not the ignitor circuit part.
The resistor values used are for particular reasons such as letting enough current to the speaker so its loud enough, and limiting the current to the transistor so it turns on.