To influence it what you need is a term, short-tailed or long-tailed it does not matter, that has at least some CTR.  It doesn't need to be large, 10 a month is more than enough. Then, with this CTR in place, the auto-suggest is influenced by sheer search volumes.  If you track your it you'll see that it takes about between a day and two days for the searches to filter through to auto-suggest. It's pretty much in-line with Insights.

If a keyword has a good history then its more likely to stay in the auto-suggest even if another term trends in the shor-term.  To influence it use social media to get people to search or run competitions etc to suggest people to search.
Pure search volume without CTR will not work. You could fire 10 million searches a day for two weeks it won't appear if there is no CTR.

Ranking above Wikipedia is the same as ranking about any other site.  You just need to be relevant with strong links and social social signals.  I have sites that rank above wikipedia on several topics because the sites are more relevant than the one Wikipedia page, but it really depends on the subject matter as to how easy it is.  But, once again, just treat it like any other site.