It happened again via. the code_editor.php page, so can't be Facebook Connect.

I want to believe that this is my server doing this, but it only happens on lovinity.org . It has yet to happen on my alfa.lovinity.org domain, which has a separate 10RC13 Composr installation (website WIP for my college radio station)

code_editor.php is as lightweight as it gets, it doesn't even involve a database connection, and the only intensive thing it has is a filesystem scan to find files to edit.

I believe you have both cloudflare and nginx as reverse proxies in front of Apache, so most likely the complexity of this configuration is leading to complex issues. A gateway timeout comes from a reverse proxy, obscuring any error behind it (if there is one). It could be the queueing behaviour of these products, and therefore some issue somewhere else entirely.

You'd need to be able to do intensive diagnostics in a complex configuration like this, checking server load, queued processes at each server level, free memory, process counts, CPU usage, blocking IO, MySQL processlist, etc etc.

I wouldn't advise anyone create a configuration more complex than is truly needed, and with complexity requires a lot of sysadmin experience. Particularly I doubt nginx is needed, as there's already static caching and DOS prevention in Composr which are the usual 'selling points' of putting nginx in front of Apache.