Steven Zahm@ Brian
Well, where to start…
Connections is a plugin for WordPress, so you would need a WordPress installation and then choose a theme of your liking many, many themes include sliders.
Your site reminds me a lot of this site: http://sscit.org/
They are using a Divi theme with Connections. The templates in use for Connections are custom, so is the search (which I did for them on contract).
That said… yes I believe you could fairly closely mimic your current setup with WordPress, a theme and Connections with some custom developed features and save money. One area Connections may not meet your needs is, presently there is no way to take payments for adding a directory entry but even that could be solved if it is needed.
If you are looking for a turn key solution to replace your current directory, I’m sorry to say that I do not think Connections would be a good fit.
Not sure if I answered your question. I hope I did and it helps.