Do search engines care if I use relative paths in my internal links instead of absolute paths? <a href="/">My website root</a> <a href="http://mysite.com/">My website root</a>
Not really but i'm not 100% sure because the google bing don't tell what the algorithm is they keep it secret.
It's all the same in the juice you pass in the url you redirected as long as you will add "nofollow" tag in the url..