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.

Setup a subdomain blog. on your site and a /blog directory and then use SEOmoz competitive link analysis. You will see it shows different DA, Google sees these as completely separate entities.

I do believe there is some value passed on, but from my own experience /blog ALWAYS wins over blog. and I've done numerous tests on this.

If I were you I would most probably nofollow those links in the footer. It also depends on how you link them, I would avoid any exact match anchor text that links the sites in the footer as this does not provide any value and Google is known to target this method.

If it were brand sites say on different cTLDs like .co.uk, .com, .ca etc. then it would be ok to not use the nofollow, however as you mention they are largely unrelated. Cross-linking within the footer (sitewide) without a nofollow may end up looking like a link wheel or network.

If there is relevance in having those links, keep them but nofollow them and avoid using exact match anchors.

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.

Chenzo is correct, install the SEO Ultimate plugin as this does wonders for deep editing of titles and descriptions for categories/tags and more. There is a setting that you can rewrite the URL from categories in the url to a specific post url.

I usually avoid having categories in the URL due to the issue you describe above of having multiple categories, I use a rewrite to just have a url such as /blog/post-name/ instead of /blog/category/tag/post-name/.