Why are equalizers moving average model (FIR) Let the equalizer coefficients obtained, say $w$ and the input to the equalizer is $y$ that is a noise corrupted received signal, $rx$ then the equalized signal, which is the output of the FIR based equalizer, should be equal to the transmitted signal / source signal. This is the working principle. But, we often perform the operation : 

equalizer_output = filter(w,1,rx).


Question 1: Most equalizers are modeled as FIR filter. My Question is why? I  do not understand how this operation gives the input or how is this performing deconvolution?

Question 2:  For example considering a single input single output channel, $x$ is a BPSK modulated source signal and there is multipath fading with 3 taps (not Rayleigh), then the transmitted signal is 
$$y(t) = h(1).x(t-1) + h(2).x(t-2) + h(3).x(t-3) + x(t)+ w(t)$$ where $w$ is the Additive White Gaussian measurement / channel noise. The above model is FIR model and not any filter. This is not the equalizer. In order to devise an equalizer inorder to get the input $x$, do we again devise a FIR filter as an equalizer whose input would be $y$? $h$ is the channel impulse response but it is not any model as such.  

Question 3: When do we model a channel as an IIR model and then the equalizer would be an IIR or FIR filter?
The reason why almost all linear adaptive equalizers are implemented as FIR filters is that FIR filters are always stable and that there exist relatively simple and effective adaptation algorithms. Note that much work has been done on adaptive IIR filters (e.g., this book by Phillip Regalia), but in practice FIR filters are still the preferred option.

Note that no real-world channel can be ideally equalized by an FIR filter, but that's not as bad as it may seem because an FIR filter can approximate the ideal equalizer reasonably well if its filter length is chosen appropriately and if the adaptation algorithm is tuned to the application. Also, perfect equalization of the channel is not always desired if the channel is not only dispersive (i.e., acts a filter) but if it also introduces noise. In this case a combined criterion is used that takes into account the ISI due to the dispersive channel as well as the degradation due to the noise.