Your meter is correct. With a 3.7 V battery and a 10 ohm resistor, Ohm's law (\$V = I\cdot R\$) tells you that the current should be 0.37 A.
The 1260 mA you mentioned is probably 1260 mAh (milliamp hours). This tells you the capacity of the battery, not the current draw. For example, if you drew 100 mA from the battery continuously, it would last 1260 mAh / 100 mA = 12.6 hours.
Is it possible to have a 5 meter wire that has 10,000 amps or more current flowing?
Yes, however these are usually referred to as "bus bars" and not wires. They are essentially bars or rods of copper (or another conductor if required) that carry the current.
I assume this can be done with a step-down transformer and the voltage could be 2V and lower
You would use a step down transformer if your power source was higher voltage and lower current than what you need. Since you haven't specified where you're getting this power from we can't recommend anything.
I assume that most of these wires have low resistance? So that heat would be considered.
Yes. The lower resistance the better. Often copper is used for this, but in some power installations they do use superconductors. That's pretty tricky due to the requirement for cooling.
You can use several bus bars in parallel to permit greater air flow, reducing your need for more metal to carry the same current, or the need for active cooling.
10kA is going to require a lot of copper, even over 5m, and copper is expensive so you'll want to carefully balance the use of the system as a whole against how much copper and cooling you may need.
Best Answer
I'd get a precision 0.01 ohm power resistor and measure the voltage across that. At 20 amps you'd only have to dissipate 4 watts at the resistor.
Alternately, you could find yourself a Hall Effect sensor. The resistor is simpler though.
Power resistors usually look like this:
Or this: