Mathematical operations with given discrete frequency response function
Is it just element-wise multiplication of the FFT array with the FRF
  array, followed by an IFFT


No. Filtering in the frequency domain is not trivial since it's equivalent to circular convolution (not linear) and will result in time domain aliasing. Google "overlap-add" or "overlap save" for a description of adequate algorithms


  To take it one step further, what if I only had a filtered signal y(k) and my discrete FRF, could I compute the original signal x(k)?


Sometimes. In essence what you are trying to do is an "inverse filter". This is only possible if the filter itself is invertible. In other words: information that's been lost in the filtering process cannot be recovered.  In practice that means you can't have any zero (or too small) amplitude in your filter and you need decent signal to noise at all relevant frequencies