1. The search in Connections is a key word search. Any entries containing any of the entered keywords will be returned as a result. You can test this in the admin on the Connections : Manage admin page.
As far as customization, there is none beyond the settings on the Connections : Settings admin page under the Search tab. These settings basically limit which fields are searched when a user performs a search.
2. There is no support for custom fields at this time.
3. Sorry, no.
I hope that answers your questions.