When is one active and one dummy strain gauge preferred to two active strain gauges?
A few points I'd like to bring up based on the link to this pdf from Kyowa Electronic Instruments: 


A dummy strain gauge can be used to negate the effect of strains you would not like to measure, like placing one on the opposite side to negate the effect of bending strain, so it's never really a "dummy".
2 Active strain gauges (config. 5-9) can provide either temperature compensation or 2x sensitivity, as far my understanding goes. The only way to have both with only 2 gauges, is when measuring bending or twisting strain. It is impossible to do with compressive strain.
2 Active and 2 Passive strain gauges form a full Wheatstone bridge, which is a very common setup to use when temperature compensation and increased sensitivity are desired. It isn't as popular because it is more expensive.
4 active strain gauges again is not really possible (with compressive strain) if you interpret the number of active strain gauges as the multiplying factor of the sensitivity (which seems to be the way you are defining the "activeness" of a strain gauge). This is again possible with bending (config 10) or twisting (config 14) strain, where you can have temperature compensation and 4x the sensitivity.


The reason why this works is pretty intuitive: With bending or twisting, different sections of your material are under opposite loads, hence you can have twice the sensitivity by measuring the difference in loads instead of the absolute difference. This is not possible with compressive loads, since the material is under unidirectional stress everywhere. Doubling the sensitivity with compressive stress requires twice as many strain gauges on opposite sides of the Wheatstone, which does not offer temperature compensation. This is because temperature compensation is achieved by having 2 strain gauges on the same side of the Wheatstone bridge. So any configuration that allows that, whether the strain gauge is "active" or "passive" will give temperature compensation