You probably want an operational amplifier (opamp) like the MCP601 connected to the two ends of the igniter, with the igniter being tied to ground.
Provide 5V out from an Arduino pin through a small signal diode, to a current limiting resistor (470 Ohm, say) to the igniter circuit.
The opamp has "+" on the output end of the resistor. Set the feedback resistors for 200x amplification. IIRC, this means 199 kOhm between out and - pin, and 1 kOhm between - pin and ground. Connect firing resistor ground and opamp ground together.
Now, the output of the opamp will be on the order of 1.4 Volts for the small igniter, and 3.2 Volts for the bigger one. You can easily sense this with an analog in on the Arduino, while providing 5V on the digital output pin.
If you get something over 4V (really, close to 5V,) then you know the circuit is open (no igniter present).
If you get something under 0.8V (really, close to 0V,) then you know the circuit is short.
The diode is there to protect the pin when you supply the 19V.
To make sound, hook up a buzzer of some sort. Radio Shack has a 5V capable buzzer that draws 30 mA or less, which you can drive directly from a digital out.
Or you can use an "analog" (PWM) output hooked to a transistor that drives a speaker. Or a cheap computer/ipod power amplifier through a 4:1 resistor divider (line in is about 1V p-p.) So, PWM out -> 4.7 kOhm resistor, tap this to tip of 3.5mm plug as well as 1 kOhm resistor to ground, connect ring of 3.5mm plug to same ground.
To control current, you need to modulate the output of the battery with PWM, and filter it with a large capacitor, to get the appropriate voltage. Or you can assume that anything above 4A is fine, and just plug the battery straight in. That will draw > 20A for the little one, so hopefully that will not kill it.
To modulate the voltage with PWM, you can't use a relay; instead, use a motor controller rated for sufficient amps and volts. The Pololu VNH5019 driver might do the trick: http://www.pololu.com/catalog/product/1451
Or, if you like margins of safety, this: http://www.pololu.com/catalog/product/1457
Note that the bigger Pololu board also has a current sensor you can use to measure the current, if you want to dynamically adjust the duty cycle.
For double safety, use the relay to disconnect the battery from the motor controller VMot in. So, the firing sequence would be:
1) Set appropriate PWM duty cycle on motor controller (connected to an "analog" PWM out)
2) initiate the countdown
3) when ready, turn on the relay so the motor controller gets voltage
Also, note that the relay coil probably is too heavy for the Arduino to drive directly (at most 30 mA or so per digital pin!) Instead, you may need a transistor to drive the coil. I recommend a 2N7000 / BS170. Hook Source to ground, Drain to coil, Gate to the digital output of the Arduino, and top of coil to coil voltage (5 or 12 V?)