How to switch between 2 LEDs using a powered momentary DPDTswitch

Both the LED (with a series resistor!) and a buzzer are viable options.

For the LED put a resistor in series with the LED. For 12V and 20 mA the value would eb 600 Ohm, I suggest you start with a 1k resistor. Do add a suppressor diode across the relais coil, any 1N400x will do.

For a 12V buzzer no series resistor is needed, but I would still add the suppressor diode. The weird effect you observed might be due to wiring the buzzer between the output and ground (instead of between output an +12V).

Believe it or not, that black lump is epoxy covering an integrated circuit, probably a small microprocessor. These are often caseless chips, just the IC die glued to the PCB with fine wires between the pads on the chip and PCB.

Many microcontrollers can be put into a deep-sleep state where they draw just a few microamps, so little so as not to significantly affect battery life. They spend all of their time in deep sleep until someone presses the button, which is connected to an interrupt line to wake it up.