Hate to be donny downer but a few issues: you actually don't need the mini contactor on the main board. It would have the same effect (and be easier) to connect the ground of the 2nd wall wart to system ground and the +5V lead to the top of a resistive divider that reduces the 5V to 3V (a little less than 3.3 for safety) and feed that to the digital input pin. When power is on the wall wart provides 5V and the digital input pin sees a high level (power on). If the power fails, the 5v drops to 0 and the digital input pin goes low indicating power failure. If you want to keep the contactor you don't need the resistors to the wall wart. If the coil is connected directly across the 5V input from the wall wart it will take 30ma of current (the wall wart can supply up to an amp, but will only supply what the circuit wants, in this case 30ma. You should connect a 1n4001 diode in reverse (i.e. cathode to the 5V pin) across the coil / wall wart to absorb the inductive spike from the relay coil though. Without the relay a 100 ohm resistor one end on ground and the other on the digital input pin of the micro with a 68 ohm resistor to the +5V connection on the wall wart will draw about 29ma and supply 3v (with 5V in to the top of the 68 ohm resistor) to the digital input pin, enough to register as high (>2.4V) and lower than the 3.3V input limit. The one driving the buzzer is more of a problem as the data sheet says the 3V coil version draws 50ma and the output pin of an arduino can sink a maximum of 40ma on one pin and about 200 ma for the entire chip, so you would need a driver on the pin to drive the contactor. A npn transistor such as the 2n2222 (which will sink 600ma max) will do nicely a series resistor of about 330 ohms from the digital output to the base of the transistor should provide enough base current to drive the relay with ease. As well the contactor coil is an inductor and when it switches off it will increase the voltage in an attempt to keep the current flowing so you need a diode in reverse across the coil terminals to protect the driver from the inductive spike. If you aren't driving an existing buzzer with the relay contacts, it would be easier to use a piezo buzzer such as the one in core parts that should run fine from a 3.3v output with perhaps a series resistor to reduce the sound level if needed. Then we hit your remote sensor 
, they all need the power supply bypass capacitors (a .1 or higher ceramic and a 10 uf or so tantalum) to insure the 5V remains 5v after all that wire. The temp sensor is likely to have noise problems with an analog signal in that much wire. You would be better off using something like a DS18B20 one wire temp module so that the temperature comes across the wire as a digital signal rather than analog. An example of using one is here:
I expect someone must make a module (and indeed ebay has a bunch) for this if you don't want to wire a chip (and I wouldn't. I'd rather buy one mounted in a probe already ).
Peter