I’m REALLY surprised to see a provider as big as GoDaddy not having persistent connections enabled. It’s possible that there’s something in your .htaccess that could be breaking it (if you’re willing to post or send your .htaccess we can take a look). As Sajal mentioned, this should be the top priority and will likely cut your page load time in half from the looks of things.
Whatever is going on with the double redirect is VERY bizarre. From looking at the headers, it doesn’t look like it is setting a session cookie and the request headers are identical for both requests for / so it must be doing something on the back-end. Do you have any enhanced security modules or options enabled that might be GoDaddy specific? The only thing I can think of is that it is some sort of bot protection that relies on the redirects being followed and they do some form of IP tracking.
I’m not sure if they’ve made it available across the board but GoDaddy has mod_pagespeed installed and available as an option that you can turn on for sites - it might be worth turning it on and making sure nothing breaks to see what it can do to help (mod_pagespeed is an apache module from Google that automatically performs some optimizations). It looks like your caching is already working well so the main benefit it would probably offer would be inlining small images and combining the js/css files.