Does a bitonic sorter compare a pair of elements more than once?
1) yes, the exact same comparison is made, interleaved with other operations. The butterfly network creates tones and checks local and far pairs, after far pairs it still has to apply local ones.

2) The original bitonic sort by Batcher requires the sequence that is a power of two. It makes tones in bitonic sort phase and then bitonic merge phase. The given algorithm introduced dummy nodes (infinite value or maximum that can be stored) for inputs $2^k - c$ and after sorting strips them from result. So the original network uses the number of comparisons equal to the net sorting the next power of two input size. The modifications that use less comparisons are not successful because of environment they run in. While it is possible to avoid not needed comparisons, it introduces branching at higher level (or special blocks, since it is no longer symmetric), to choose what comparator should be used and (in my opinion) is not any more bitonic sort.

You might be interested in more general sorting networks, where the exact number of compares is the subject of creating the sorting net for given $n$. The structure and algorithm depends on number of comparators available. The net is created for exactly given size, and it is not required to be power of two or be even. The optimal networks are found for $n \le 10$.

I recommend Optimal sorting networks by Daniel Bundala and Jakub Závodný about pursuit of better networks (these are general, not bitonic) with algorithmic approach.