Just a note that 360works does not show support for FMS13 for Web Services Manager. I know they're probably hard at work building new releases for their massive product line.
As far as this behavior, I recall a similar thing happening when IWP was enabled on windows servers in previous versions. Unfortunately, with Filemaker's strict dependence on port 80 as part of their port simplification, you're most likely going to have to do a hackjob to your machine's network stack to (attempt) to make it work.
I'm not terribly familiar with the mac server side, and definitely not port splitting/sharing. Sorry I couldn't be more help.
Web Services Manager will indeed work with FileMaker Server 13. You'll need to install it slightly differently, as you're right, 13 redoes their web server setup, but we can still piggyback off the new way. Server 13 uses either a new copy of Apache, or a new Web Site in IIS. You can still place pages the need to be served within their new setup. To install, simply copy the php file instead to: FileMaker Server/HTTPServer/htdocs/ and everything else should work the same. If you run into any issues, please let us know and we'd be happy to help!