Which assumption to consider first when there is more than one diode
You simply have to consider all possible cases, and choose the one that doesn't lead to a logical contradiction: 

Case 1: Both diodes off. Then point "V" is at -3 V, and the voltage across D1 is 4 V and the voltage across D2 is 6 V, contradicting the assumption that the diodes are off. So we reject this case.

Case 2: D1 is on, D2 is off. Then the voltage at "V" is about 0.3 V, and there is 2.7 V across D2, again contradicting the assumption that D2 is off. So we reject this case.

Case 3: D1 and D2 are both on. Then the voltage at "V" is both 0.3 and 2.3 V at the same time, again a contradiction, so we reject this case.

That leaves the correct solution: D2 is on and D1 is off, which indeed produces no contradiction and leads to the solution of the circuit.