Why is the F-measure preferred for classification tasks?
The Fı-score is preferred to simple classification accuracy in order to counter the problem of imbalanced datasets; if the thing you are looking for occurs only rarely anyway then a naive classifier can always say no and appear to be working very well! A variant on Fı is Fß, where 

Fß = (1+ß²) &times; [ (P &times; R) &divide; ( (ß² &times; P) + R ) ]

Vary ß to balance precision and recall. As to the why F or G, I believe it to be empirical - you don't say if you are classifying or clustering in your own application?