There is no better method they do not affect rankings, it is purely personal preference. However you must implement proper redirect rules to resolve http://mysite.com to http://www.mysite.com or vice versa which ever one you choose.

I tend to always go for www. as it just looks better to me.

A banner ad can be a legitimate link source, it will be an arrangement between the site owners and not done through an ad network. It would just be like you coming to me asking for a link on my site, but instead of anchor text using an image. Not using a no-follow tag is against guidelines, but you could argue it's not a banner ad and a legitimate link.

Link juice is definately not passed through banner ads on ad networks such as adwords/doubleclick or affiliate banners as they bounce over an adserver first.

If the text is exactly the same in each article then yes, Google looks for large chunks of duplicate text. Usually the way to do this would be to rewrite the article for each site.

domain.com/blog is current best practice for hosting a blog to benefit your domain from content freshness and linking. A subdomain will be seen as a seperate site.

There are commerical reasons why you would want an external blog such as for customer service issues or a branding exercise, but if you want the benefits of the links and content a blog will generate host it on your domain root.