Doubt on Short Circuiting
The resistance can be negligible for one purpose (figuring out the current drawn from the battery), but not negligible for another purpose (figuring out how much the wire heats up).

In the first case, the wire resistance becomes negligible when it is "much less than" the internal resistance of the battery.

In the second case, the wire resistance isn't negligible because you're considering an effect where some other resistance will dominate the behavior. When you're calculating the self-heating of a resistor (or wire), having already determined the current, the only resistance that matters is the resistance of that wire itself.