I put the two probes together to test if the fuse is blown, but it gives me a result of 0.3~0.7ohms, so it should be okay.
On a good meter, the fuse is only in the path for current measurements. If the fuse is blown you can still measure resistance & voltage.
Try measuring current for an LED & resistor. If the LED doesn't light, the meter is "open".
Blown current fuses are very common.... All you have to do is accidentally switch the meter to current with the probes across a power supply or battery, and "poof". Better meters usually have a separate banana jack for current measurement, so you have to move the leads to measure current. That way, you are less likely to make a mistake.
Does the current usually flow through even if the digital multimeter is switched off/measuring voltage?
When it's off, who knows???
When measuring voltage the meter's resistance is very high, meghoms. Yes. Current flows through the meter, but it's a tiny amount of current (Ohm's Law). The idea is that a high resistance in parallel doesn't interfere with the circuit under test.
When measuring current the meter's resistance is very low. This means that there is very little voltage drop across the meter and the low resistance in series doesn't interfere with the circuit under test.
When measuring resistance, the meter puts-out a small voltage & current. As you probably know, you can't measure resistance with the circuit-powered-up because that messes-up the resistance reading. In the old days, an analog meter could measure voltage & current without a battery, but if the battery was dead you couldn't measure resistance.
If you put the (volt) meter in series with your circuit you are creating a voltage divider. In series with a "normal" load like the Arduino, most of the voltage will be dropped across the meter (the higher-resistance side of the voltage divider). You'll measure the full power supply voltage but the Arduino won't be powered-up because there is very little voltage across it.
In current mode, the meter has very-low resistance.
If you have some low-value resistors, you can use a voltmeter to measure current (Ohm's Law). That's how your meter works anyway.... For example, put a 1-Ohm resistor in series and if you measure 200mV, you have 200mA.
The trick is finding a resistor that works for the current you're trying to measure and the sensitivity/resolution of your meter... If the resistor value is too low (and/or the current is too low), the voltage drop will be too low to measure. If the resistance is too high (and/or the current is too high), you'll get too much voltage drop across the resistor and your circuit won't operate correctly.
