It's usually a bit hard to debug things like this, but from the redirect urls your webserver is giving back I think it's your web server rewrites that are the issue. I think with your current configuration Magento is confused between using
/index.php/ including urls and not using them.
It's possible that following this stackoverflow answer will help you solve that issue (and likely more, incl. SEO ones):
I should also note, although a bit off-topic for the specific answer, that while you are also running a rather unpatched version of magento which exposes you to security risks, you are also running a misconfigured Apache server that exposes your var folder in a way where it makes you susceptible to the cacheleak vulnerability.
I hope this resolves your issue.