Unfortunately, the site is within our network. There’s no way to provide outside access. I can post some screenshots, though, if that would help.
That said, I’ve solved one of my problems. I disabled all plugin on my site (except for Connections-related plugins) and the drop down Category list began working again. I slowly re-activated the plugins until I found one (404 Redirected) that was causing the problem. (Which makes sense in hindsight since this plugin grabs URLs that WordPress doesn’t recognize… which includes the URLs generated by Connections for the category drop down.)
Unfortunately, search is still not working. Disabling that plugin has given me some progress. The URL is now displaying properly. Searching for “hospital” now gives me “/services-directory/?cn-s=hospital&cn-cat=” instead of just “/services-directory/” (the page that this is on). However, the results include all items, not just items with “hospital” in the name. Any ideas for debugging this?