Arduino and resistors
Your relay needs 12 V / 400 Ohms = 30 mA of current in the coil. The transistor you are using has a DC current gain of at worst 110 (from the datasheet). So you need at least 0.27 mA going into the base. The Arduino puts out 5 V and the transistor base is at 0.7 V, so the maximum resistor you can use is (5V-0.7V)/(0.27mA) = 15.8 kOhm. Any larger and you will be limiting the current available to the coil. 10 kOhm is a nice round value that is sufficient. You could use smaller, but then you'd be wasting power on the base current.


The page linked says you need hFE at least 5 times the ratio of collector and base current. This is crazy. Yes, you need some design margin, but there's nothing special about 5x and it seems like overkill. 


Because that's how that particular sensor has to be wired, check the datasheet.
Again, read the datasheet for the parts in question. As for a shielded cable, that seems like overkill unless you are in a seriously electrically noisy environment (like a factory). Try it without and if the measurement seems noisy try putting a capacitor to ground at the input pin of the Arduino. You might also want a series resistor to form a low-pass RC filter.