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/.

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.

You need to add a redirect preferably in the .htaccess file so that index.php automatically redirects to the root. This should fix the problem as Google will currently see www.yourdomain.com/index.php and www.yourdomain.com/ and www.yourdomain.com asa 3 separate pages thus why your page authority is different. Google will also think they are duplicate content which is obviously bad!

This fix will work if mod_rewrite is enabled, but is usually enabled on most apache servers. http://www.askapache.com/htaccess/redirect-index-blog-root.html

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.

Google see the text anyway as it is hard coded into the html, the JS is only used on the front end to show the user so they can hide/show the text.