Hey Matt,
Lithium's "Billable API calls" refer to any API calls made from an external party to a Lithium community.
1) Since Coveo is an external party, you would have to check with them about how many API calls they make to Lithium and when. If I recall correctly, I believe they used to use a crawler behind the scenes and then store the results which would not result in API calls getting incremented, but it's best to check with them.
Unfortunately, I'm not too familiar with our SFDC connector functionality, so I would have to defer to VimalGa for the other 2 questions.
Hope this helps,
Yuri