I was able to get this resolved. Bottom line was that IIS did not have everything installed that it needed. I logged into another server (running 2008 R2) and compared the services that were running on that instance of IIS to what had been installed on the 2012 box, and found several things missing.
When I read the post above from Sam71, I assumed that he had installed .net 3.5 only to get the installation checker to run, but that was likely the reason that the webpages wouldn't load as well.
What I ended up doing was uninstalling 3CX (again), adding ASP.net (for both 3.5 and 4.5), and then reinstalled 3CX, and almost everything worked. I was still getting an error on the management webpage. I went into IIS and found that "Modules" under "Feature Delegation" was set to "Read Only" for the 3CX website. After changing that to "Read/Write", all pages appear to be functioning properly.