I don’t really have any public available church examples that I can link you too because churches tend to keep their member directories private to respect the privacy of their members. So I’ll try to answer you questions the best I can…
Well, there are no print option available. So you would not really gain anything their by switching to Connections.
Can more than one family be listed on a page? Well, yes. If you look at any of the examples, they all list more than one entry per page, a family would work the same way.
Through configuration you could achieve a page with a few details that link to a page with more details. There’s a couple methods you could use to achieve this with Connections. The first would to use a mixture of one of the premium templates with one of the core templates. For example use Slim Plus of Tile Plus for your less detail and then use the Card or the Bio Entry card template for the more details. Another way, would be to setup two pages. One page for the list and another for the result/detail. You would need one of the premium templates for this work effectively.
If you decide to use Connections, I can give more thorough directions for both methods.
Hope that helps!