Thanks for your interest in Connections. I’ll try to answer you questions as best I can…
1. You will have to select a Directory Home page, especially if you’re going to use Connections on a post. The reason is that all links Connections creates are based on and resolve to that directory home page. Using Connections on a page is different because all links within the entries will be based on the current page and resolve to the current page. This is actually to support multiple directories; split based on categories for example. The one notable exception to this rule is that the Widget Pack links will always be based on and resolve to the directory home page.
2. Yes, you would add a visibility column to the CSV and then define each row as public, private, or unlisted.
3. The relation links only show when the entry is a Family, there really is not a way to hide this because that is part of the template. Options would include hiding it with CSS or manually editing the template’s PHP file to remove it.
4. The only option here is to manually edit the template’s PHP file. Which template are you currently using where the name is below the photo? I’m tired at the moment but I’m pretty certain one doesn’t exist with that layout. Or, maybe I don’t understand, sorry.
I hope that answers the questions thoroughly!