Voltage across a resistance controls current in that resistance. It's that simple. Ohm's law, I = E/R. Double voltage across a resistance, you double current through it.
Note, this is an ideal answer, in practice things change a bit, because resistance is not just confined to a fixed resistor in a circuit.
Connecting wires also have resistance, and batteries have "internal resistance", usually considered as a fixed resistor in series with the battery.
Any given battery will only supply a maximum current depending on size and type, so what happens is that IF more current is expected of it, its TERMINAL VOLTAGE will fall to match the voltage to the current max. it can supply. That's that "internal resistance" at work, the voltage "drop" (lost) is across the "internal resistance". Who really cares about why? You choose the most appropriate TYPE, and terminal voltage, of a battery for the particular application.
Yes, power increases exponentially.
High current, low voltage, high voltage, low current can both be the same power, because power by Watt's rule is voltage x current. So 100V. at 10A. is 1,000W., and 10V. at 100A. is 1,000W. Same power.
I think you are overthinking too much, and confusing yourself.