Where in the signal chain is Channel Equalization applied?
Generally, any matched filtering would be done before equalization, since there is no practical advantage that I am aware of to applying a matched filter afterwards.  Most textbooks that I've read present an architecture in which a matched filter is applied first, the signal is sampled at 1 sample/symbol, then an equalizer is applied at 1 sample/symbol.  But it does not have to be this way: you could apply a matched filter, but not downsample, then apply a fractionally-spaced equalizer and sample the output at 1 sample/symbol; or your could skip the matched filter entirely and apply an equalizer directly, in which case the equalizer would adapt to include both the matched filter and equalizer response.

Regarding fractional delays: if the equalizer is symbol-spaced, then we must perform symbol timing recovery prior to downsampling so that we sample the matched filter output at the correct times.  We need to perform symbol timing recovery anyway to correct any rate offsets, but if we are downsampling to 1 sample/symbol, we must also get the symbol clock phase correct.  If the equalizer is fractionally-spaced, then it can adaptively refine the sample time, although it can't correct any substantial rate offsets.  This is an advantage of using a fractionally-spaced equalizer; the primary disadvantage is the higher input sampling rate.