Deconvolution Using Complex Division in The Frequency Domain
It's not about the index, it is about the Filter :-).
Think of your coefficients as filters and what would happen to a Sine Signal with frequency of $ \frac{N}{2} {F}_{s} $ that would be filtered (Convolved) with your samples of signal $ a $.
You will sum samples with the same absolute value yet one is negative and the other is positive, namely their sum is zero which means nothing will be left from this signal.
In practice we never use complex division for Deconvolution.
The minimum you can do is use some kind of regularization.
Zeros in Frequency Domain
In the noiseless case, having zero magnitude at some indices means those indices were zero either in the input signal or the filter.
The problem is we can never know hence when we want to recover any of them the data is lost. This is due to convolution being multiplication in frequency domain so at some index $ {C}_{k} = {A}_{k} {B}_{k} $ and if either $ {A}_{k} $ or $ {B}_{k} $ is zero then $ {C}_{k} $ is zero without being able to recover $ {A}_{k} $ or $ {B}_{k} $.
It means, that, potentially if there are no zeros for the data or the filter (At least in their overlapping domain) data can be restored.
In practice, this is very hard problem given noise.