The question that has always struck me as central, when I was a student and ever since, is this:
(1) In what ways do languages differ and in what ways are all human languages alike?
When I was a student it was common for answers to this question to be proposed based on evidence from English alone. In my dissertation and ever since I have tried to enlarge the language base in terms of which this question is discussed. One chapter of my dissertation, in fact, later evolved into a much-discussed parameter of variation.
What has happened to question (1) since my student days is nothing short of amazing. There has been an explosion of research on the most diverse languages, largely due to the development of theoretical constructs capable of handling a far wider range of languages than the constructs in use in my student days could handle. The advances in understanding, I think, have been impressive, although the splintering of the research community along theoretical lines, at least in syntax, has to a considerable extent obscured the real gains that have been made.
The question itself continues to be central to linguistics. It has not been answered not because it is an ill-conceived question, but because, I would say, it is right on the mark. The results of research on typologically diverse languages have brought out greater cross-linguistic differences than were even imagined in my student days. To cite just one example, my own expansion of my research to sign languages alongside spoken ones has made me keenly aware of the possibility of far greater cross-linguistic variation than previously imagined and has made finding cross-linguistic commonalities much more challenging.
From my own perspective, I think I know far more about how languages work than anyone knew in my student days, but at the same time, what I think remains to be discovered amounts to much more than anyone imagined in my student days. And that, I think, is real progress.