Does the lane 3 second count start when the ball is given to offensive player taking the ball out of bounds or when the ball is thrown in bounds by the offensive player?

My best guess here is that you really mean the 10 second backcourt count. It begins when there is team control on the floor. Team control is established when a player establishes player control. Player control is defined as a player holding or dribbling a ball inbounds.

So, if you really mean the 10 second backcourt count, it starts when an offensive player on the floor receives (not just touches, as in a deflection) the ball and is in that team's backcourt.

Mark -- The ball could be coming in under team A's (offensive) basket, rather than backcourt.

No.

According to the rulebook, the 3-second count doesn't start until there is team control by the offensive team in the frontcourt.

(In HS, there's no team control during a throw-in.)

According to most officials, however, the 3-second count doesn't start until hell freezes over. :)

3 second count starts when the rookie official calls a state championship game.

How does that pertain to what I said?

Well, if the ball comes in under A's basket, of course, the 10 second count isn't an issue, but the 3 second count is. Am I missing something here?

Getting back to the original question:

Answer:

Neither. The 3 second count will begin when any player from a team is standing in the lane (including the lines) and his team has control of the ball in it's frontcourt.
The count continues during an interrupted dribble.

So, on a throw-in the only count is the 5 second count to inbound the ball.

Depends on where the throw-in goes! It's not where it starts that matters.