Help in problem formulation for estimation of image as a feature vector - SISO or MIMO FIR channel model?
This question is confusing, but I thought I'd try to make some sense of it.

Suppose you are transmitting one of your $\mathbf{x}_i$ vectors one component at a time. That is, the transmitted signal, $s[k]$,  for the $i^{\rm th}$ feature vector is:
$$
s_i[k] = x_{ik}
$$
If this signal is corrupted by a channel with impulse response $h[k]$ then the received signal, $y[k]$ will be
$$
y[k] = h[k] \star s_i[k] = \sum_{m=0}^{M-1} h[m] s_i[k-m] = \sum_{m=0}^{M-1} h[m] x_{im}
$$
where $\star$ represents convolution.

If we use this formulation, then the answer to:


  Should the source input be each component of the feature vector or an entire example?


is that it should be each component of the feature vector.


  if the input in the form of the feature vector is transmitted via a channel whose impulse response is modeled as moving average (finite response, FIR) then inputs and how many outputs should there be?


There should be a sequence of outputs. If the FIR filter is length $M$, then the output will be of length $M+d-1$.


  Would the channel be Single input-Single Output FIR or Multiple Input Multiple Output FIR ? 


In the model above, it is a SISO system.


  Would the estimation be performed using one example or all the examples?


If the channel does not change, then you might be better off using all the data you have available.


  Should there be $d$ channel coefficients?


There is no point in tying the number of feature vector components with the length of the channel. In the above, the length of the channel is $M$.