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Never take down the page. Put a text above all "Hey this contest is closed!" if you destroy the page you loose all the trust you gained, don't do this. I my opinion a simple thing si to put some link from the page contest to the page you want rank for 
I'm italian so sorry for my bad english... I think that you may produce a few video (viral, funny or really useful) and publish them on YouTube. YouTube is great to share your brand with people. Inside the video and in the description (also in the tilte) you put your website. On your website, instead, you can publish technical/specific/vertical video which may interest only your targeted visitors.
So YouTube for spread the word about your website and your knowledge and your site for deeper content.
Don't you need SEO? I don't understand this statement, this is SEO. By the way, with "duplicate links" do you mean "duplicate contents"? Sorry but probably my english is poor (I'm italian).
Is domain.com/forum your site? If it is you probably missing the rel canonical tag, so the parametric urls create duplicate pages/title, etc... Is this a good point to start?
I don't think. Many sites uses CDN services (like Amazon S3), is it plausible that these sites get negative effects on rankings? (By the way, I'm not sure at 100%). Probably other 'mozers' can go deeply in the question.