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Yes it will be better as a sub-domain will be treated separately.  Where is the forum currently located though, on a sub domain? If so make sure you keep the old URLs active with canonical tags or redirects to the new locations so that users and bots get pushed over.

The downside is that if your forum is likely to be penalised for spam, then it will start to effect the ranking of your whole site rather than just the subdomain.

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.

I assume you are talking about Mod_rewrite with apache.

Apache's rewrite would be a good start http://httpd.apache.org/docs/2.0/misc/rewriteguide.html.  You say you are not technically strong, but you should be able to copy details from that and just change things around.  The matching pattern works on regular expressions so if you have every used this you will find the syntax makes sense. If you haven't then it can look daunting when you first start but its really not that bad and through trial and error you can quickly pick it up.

This is the full reference for mod_rewrite http://httpd.apache.org/docs/current/mod/mod_rewrite.html.

Alternatively once you have decided on your friendly URL to actual file mapping, post those in here and I am sure people will be able to help you create the rewrite rules

Forums are open ended so you can easily end up with pages that are low in quality content and with post-panda google this could have a negative effect on the whole domain.  Forums are also targeted for link building schemes which, unless actively moderated, could end up with you linking to lots of illegal sites that could end up penalising you.