Is power proportional to V or V^2?
It depends on the circumstances. Without knowing anything else, we don't know whether power is proportional to \$V\$, \$V^2\$, or neither.
If a variable voltage source is connected to a resistor, then the power is proportional to \$V^2\$.
If a variable voltage source is connected to a constant-current load (something which admits the same amount of current regardless of the voltage across it), then the power is proportional to \$V\$.
If a variable voltage source is connected to a constant-power load (one which admits more current when the voltage is less, and less current when the voltage is more, so as to consume a constant amount of power, and thereby acting as a negative resistance), then the power is independent of \$V\$. A switched-mode power supply may behave in a manner similar to this. It would be pretty straightforward to design a device which consumes \$1\ \mathrm{W}\$ of power when connected to any voltage between \$5\ \mathrm{V}\$ and \$24\ \mathrm{V}\$, for example.