First : Of course current flows through R3 and/or R5. They - and nothing else - connect to one end of a current source. As it is sourcing current, it must flow somewhere, and that means either R3 or R5 or both.
Second : Re-draw the circuit with nodes 1 and 2 shorted. Can you see that R1 and R4 are now in parallel? The same is true of R3 and R5. That lets you draw a much simpler circuit. You can now work out the voltage between node 3 and node 1/2 - ditto node 4.
At which point you know the voltage across every resistor, therefore the current through each. (Check your math : the R1 and R4 currents must sum to 20mA, ditto R3 and R5).
And having computed these, you can probably see how to calculate the answer to your question.
1) Yes, connecting a voltage source directly to its return with a wire creates a short circuit.
1b) Yes, discharging a battery at too high a current draw will overheat the battery and can result in a catastrophic failure (blowing up, boiling electrolyte, fire, other bad things).
2) To just heat a wire, ideally, you would connect it to a controlled (regulated) current source. Then you can adjust the available current to control the heating. Most mid-range and up bench power supplies have a current limit mode.
3) Resistors server many purposes, you will need to be a little less broad. Resistors can limit current, but at the cost of heating up. They won't "Protect" the wire actually, just reduce the current available, and make the resistor a heater.
4) A Taser generates very high voltage pulses, at a very very low current. There is a lot of circuitry between the taser contact points and the battery... Again, really too broad to answer here.
Best Answer
In simple and practical terms, a short circuit is an unwanted or unintentional path that current can take which bypasses the routes you actually want it to take.
This is normally a low resistance path between two points of differing potential.
For instance:
simulate this circuit – Schematic created using CircuitLab
In the left simple LED circuit, just over 6 mA is flowing round the circuit. Create a short circuit, represented by a very low resistance (no wire is a perfect 0 Ω conductor) and 5000 A wants to try to flow through it. That's bad news for the battery. The battery could well explode. What is certain, though, the internal resistance of the battery will limit the current that can exist and a large voltage drop will be seen at the terminals of the battery causing the whole circuit to stop functioning.