As with all such questions, it depends (on the purpose of the studies, and the type of wind generator). What is modeled should be what is there. Induction generator turbines should use an induction motor model with negative real power (so that their fault current contribution, and reactive demand increase when (re) accelerating are captured). 100% inverted turbines are usually represented as a static P (possibly Q) load. Double fed turbines...may require careful thought. Obviously, dispatch/scheduling becomes more complicated, and in general the number of study cases will be increased much more than by a comparable thermal generator. Where there are special features (such as voltage control, or associated SVCs) these also need to be modeled.
If the fault is on the generator itself (or associated equipment) then obviously it trips. However the assumption in your question is not usually correct; normal generators of significant size should not trip for other power system faults. Clearing times are kept below the critical times (for common fault types) and plant design ensures system undervoltage does not cause trip due to loss of auxiliaries.
These rules may not be applied to small dispersed generation, firstly because it is not necessary (The rules are intended to prevent faults cascading) and secondly because it is often inordinately expensive (In particular, clearing times on conventional MV distribution systems are usually too long for stability). Many (all?) early commercially available wind turbines had very poor under-voltage ride-through capability. As wind farms became larger utilities have started to impose more stringent requirements.
Although direct-connected induction generators are feasible (and exist), and synchronous generators possible in principle, they are forced to operate at close to/exactly fixed speed respectively by the power system. This means they are not efficient at capturing the wind (because efficiency is maximized at a speed that increases with increasing wind speed). Consequently, Siddhartha is correct that most large wind turbines do not use direct connected generators. They are either 100% inverted, or partly inverted via a DFIG (which allows a partial range of operating speeds). In terms of absolute numbers i am not sure whether induction machines dominate (One European manufacturer in particular made very many smaller direct connected induction generators), but in the larger sizes my impression is that it is relatively rare. Inversion also avoids the need to compensate the reactive demand of IGs, and at least in principle allows for grid voltage support.