1) API, ESTA and the I-94W form request the address of your first night's stay upon arrival into the US. As your first night is outside the US, enter in-transit to Mexico (you may need to call BA to add this to the API).

2) I never get around to updating my ESTA - sometimes when I do it is quicker to file a new application. Never encountered any difficulties at immigration.

3) When asked the purpose of my trip, I state business or pleasure. If no further questions are forthcoming, I leave it at that. I never venture to add info not requested. TBH, it is unlikely anyone will bat an eyelid at your itinerary.