I've just been fixing a few things on my daughters mini related to the cooling system
I'm pretty sure the fan draws a lot more than 5A
The radiator fan runs at two speeds, the low speed is supposed to come on at 105C, or if you have air con, when the air con is running.
The high speed fan comes on if the low speed fan doesn't cool the engine enough (112C) but I think that by the time that comes on the engine is probably very close to overheating and unless your cooling is perfect it could be too late
The low speed fan depends on a large resistor to limit the current (and therefore the speed) but it is common for it to burn out and leave no low speed fan.
I would check that the low speed fan is coming on correctly at 105C and cooling the engine down. If it comes on too late then check the thermostat for the fan. If it does not come on at all or only at full speed at 112C then check the resistor for the low speed fan has not burnt out.
Don't rely on the temperature gauge to tell you when the engine is overheating. It is controlled by the ECU and will show the rise in temp up to the normal temperature but after that will not rise until it is again very close to overheating. You can read the temperature from a OBD tool connected to the OBD port.
You can also display the temperature on the digital display by following these steps.
With the key in the ignition, but in the off position, press and hold down the odometer reset button, while holding the button down, switch the key in the ignition to position 1.
The screen will say 'Test' and a number.
Scroll through, (pressing reset button) through to test 19 and wait a moment.
Wait and the message will say "log i-off" then "log i-on" and back to "log i-off" again. When "log i-off" appears, press again.
Now you can press reset to get to 7.0 and this is your water temp in Centigrade.
If significant amounts of coolant have been added then the system needs to be bled carefully to ensure that all the air has been removed from the system, this can be tricky; there are several bleeds screws in the pipework that should be used to make sure the air is removed.