Do we see the time passing at far a way galaxies (say 12 lightyears), and thus have a very fast growing distance, more slowly?

Yes. The main evidence for this is "supernova light curves"; the rate at which the output from a supernova falls is related to its brightness in a very predictable way. From this we can measure the time dilation of distant objects. As expected, this macthes exactly with the red shift.

Does your answer means that special relativity (in combination with general relativity?) also works for the expanding universe?

Yes, the whole big bang model is built aroudn general relativity.

Thanks