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Assuming the DMM is rated to test diodes (not all are), meaning that it presents more than forward drop voltage to the diode, a shorted diode will test nearly zero ohms, usually in both directions.
Yes, but only at a basic level. If the diode measures low resistance in one direction, high resistance in the other, you know that it is not short-circuited. But... 1. If it's a voltage-regulator/reference (Zener) diode, you do not know whether it has the correct breakdown voltage, 2. If diode leakage is important, you have not tested for leakage and the ohmmeter test does not do this reliably, 3. Your ohmmeter test voltage is probably no more than 9 volts, so you have not tested for high-voltage breakdown, and 4. If it's a rectifier (especially a high-current diode) you have not tested its forward voltage at full load current.
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A: Nobody can answer that. It depends on the diode, battery on the meter, scale of the meter. It should never read zero or close to zero ohms and reversing the lead it should just be close to open but it may read some hi k ohms. A meter test is just to find shorted diodes and extremely leaking diode.
To test a diode, you can do a variety of tests. The easiest may be measuring the resistance across the leads of a diode. One side should read very high resistance, over 1 megohm, while the other should read a moderate amount of resistance, maybe a few hundred thousand ohms. This is a sign that a diode works. If both sides read very high resistance, the diode is open. If both sides read very low resistance, then the diode is shorted.