One of the questions on this years test dealt with this and brought up an interesting question (for me anyways).

1. Ball is established in front court. A1 passes to A2, but ball is deflected by b1 and ball goes to backcourt. Let's say the ball rolls all the way down to the baseline where a2 finally touches/gains controll of the ball. When does the 10 second count start? I'm thinking as soon as the ball touched the bc as team a still had team controll of the ball.

2. On the throw in after a made basket, when does the count begin? Is it when team a touches the ball? Or when team a gains control of the ball (ball is fumbled on throw in)?

Fed rules:

1. The 10 second count begins as soon as the ball gains back court status. From 9.8 "A player shall not be, nor may his/her team be, in continuous control of a ball which is in his/her backcourt for 10 seconds."

2. The count does not begin until team control is established, which is not until player control is established, which is not until the player is holding or dribbling a live ball inbounds.

A player shall not be, nor may his/her team be, in continuous control of a ball which is in his/her backcourt for 10 seconds.

That should answer both of your questions.