Q. Is there a site I can see which airlines fly where, and which offer those routes as nonstop?
A. Yes, this tool from Innovata allows you to enter any two cities and sort by nonstop. It lists Southwest and other smaller carriers that don't always show up on the radar, which we like. But it does seem a little moody, or buggy, at times, and doesn't always seem to list every possibility for international flights.
Also similar is this tool from OAG which searches for direct flights on your given dates. The same site offers other useful tools for figuring out flight duration and terminal info, as well as some that seem slightly less useful, such as the oddly titled "Who Am I Really Flying With?"