Actually, it will draw 1/10 of the current at 1/10 voltage, producing 1/100 (1%) of the original power and heat!
This is because of Ohm's law: voltage = current * resistance (E=IR). Using algebra, I=E/R. Because the voltage, E is 1/10 while resistance, R is constant, I is 1/10.
Power = voltage * current. Because both voltage and current are 1/10 of the original value, power is 1/100 of the original value.
Your heater will produce 1% of the heat at 12V. Sorry.
It depends on the level of accuracy you require.
Off the cuff, I would expect single-point voltage measurement to introduce about 10% error on power measurements at remote points.
A properly designed LV distribution system (Australia: 230 VAC single phase, 415 VAC three-phase) should keep the steady-state voltage drop within about 10%. That means there should be no more than 10% voltage difference between the supply and the load terminals. (10% isn't a mandatory limit - more of a rule of thumb.)
Phase shift doesn't seem like it would be an issue. If in doubt, look up some typical cable impedances (as found in a standard like AS/NZS 3008, or a cable manufacturer's catalog i.e. Nexans / Olex or Prysmian). You can use the cable impedances to calculate the voltage drop and phase shift for your given load current.
At high voltage (3.3 kV, 11 kV) there really isn't much voltage drop on cable runs. I would expect single-point voltage measurement to work well here. (This is, in fact, how it's done. See below.)
With regards to voltage spikes affecting your metering: don't worry about it. You are measuring power consumption (kilowatt-hours) not instantaneous power (kW). The spikes average out over time.
For perspective, voltage transients from motor starting typically disappear within 10 seconds. There are 86,400 seconds in a day. Voltage transients won't affect your power measurements.
Note: In practice, power measurements are normally taken at the upstream end of the circuit - at the switchboard/motor control centre.
For low-voltage MCC's, each meter gets a direct connection to the phase conductors. Meters can be built to measure 240 VAC directly with no problems.
For high-voltage MCC's, the voltage is too high to measure directly. A single voltage transformer is provided (i.e. 3.3kV/110V) to scale the voltages down to a measurable level. That scaled 110V voltage is then distributed to all the meters on the switchboard.
Metering isn't done at the downstream end (where the plugs are) because that's typically an exposed environment, subject to lots of abuse and damage.
Best Answer
In order to overcome these sort of sampling measurement errors you can make the problem "easier" by adding a large electrolytic capacitor on the power feed to the development board after the series measurement resistor. This is an "attempt" to measuring average current taken by the dev board. OK, I say you are measuring "average" but it depends how big your capacitor is in uF terms and how long your "violent spikes" are.
If you have a 1 ohm series resistor and a 100uF capacitor, the time constant of this is 0.1ms so you'll need more capacitance or more resistance. To get reasonably decent measurements go for a time constant of about 10ms - this should suit your multimeter.
As an aside, your multimeter may be perfectly capable of averaging over a 1ms period and giving you the results you think you may not be getting.