It's amps and volts. I can take a car battery and apply it to your skin, and nothing will happen, even though it's got tons of amps in there. The voltage is just not enough to push through the body.
Will the e-cig batteries kill you? no. they have 100-300 miliamps in them, and only about 4 volts. You'd have to stick your tongue on the end and activate the battery just to feel anything from it. (remember sticking 9v batteries on your tongue as a child?)
Nope. Wrong wrong and wrong. Because we are dealing only with a resistor, you can use the simple formula V=I*R. That is it. The heat comes from the current. The temp of a NiCr wire (neglecting heat loss from air and the evaporation of the juice) is proportional to the current passing through it.
Since the battery is about 4V and the resistor is about 4 ohms, there is 1000 mA. This is assuming the battery can generate that amount of current and there is no current limiting built into the auto batteries.
It doesn't matter how much current capacity a battery has (like a car battery). All that matters is how much resistance you put across it. Your skin is a poor conductor. At DC voltages, skin is megaohms. Even if you wet your skin or use your tongue it is still pretty high resitance. Your body has much less resistance at high higher frequencies which is why 110AC is more dangerous. (Really high voltages are another matter, that don't apply here.)
Batteries don't have "milliamps".. it IS confusing they way they label them as mAh, but that is a measure of capacity. Lithium Ion is also unique in that they can conduct current at their rating unlike NiMH. So 500mAh LiIon can put out 500mA (you don't want to suck more current out of it without hurting the battery ***Some newer chemistries like LiPo can). That battery can put out 500mA for one hour, or 250mA for 2 hrs, etc.
I could kill you easily with a eCig battery. It does have like 2-10 Watt-hrs in it. But, it would involve transforming the voltage or converting the DC to an alternating current. There is a lot of stored energy in there -- think about how much heat it produces.