>Can I just hook my laptop up to it instead of spending money on a monitor I may never use?

yes. installing an os without a monitor may be difficult, but you can probably borrow one from another computer for that...

>What's a decent OS I can use to run it without reformatting everything?

if the drives are formatted as fat32, i'd recommend freebsd, dragonfly, netbsd, openbsd, or linux (in that order). if they're ntfs, i'd recommend reformatting them.
you'll probably have to format whatever drive the os is running from, but you probably already know that.

>What can I do to make it super-quiet and guarantee as much uptime as I can possibly get?

underclock your processor enough that you don't need to use a noisy cooling system and put freebsd on it.