The short answer… it doesn’t seem Connections will provide the solution you’re looking for, sorry.
1. If info is not entered in a field, that field will simply not be displayed at all. Also, there are no custom fields.
2. Categories are it, no tags or such.
3. Would be possible, but a custom template more product focused rather than individual focused would be needed.
4. An entry can be linked to a single user, using Link, but not a group of entries.
5. You’d have to use a third party plugin; likely a membership plugin.
6. Connections operate independently of the theme, so you could use any theme.
I don’t know of another plugin that meets your requirements either, sorry. I think you might be in the roll you own custom solution territory.
Hope this helps.